Intermediate Fluid Mechanics

🧭 Overview

🧠 One-sentence thesis

This intermediate fluid mechanics course builds on fundamental concepts to teach analysis techniques for external potential flows and intermediate viscous flows, enabling students to determine forces on arbitrary bodies and predict drag in boundary layers.

📌 Key points (3–5)

• Course scope: covers external potential flows and intermediate viscous flows, with applications to aerodynamics and drag prediction.
• Prerequisites: assumes background in fluid definition, hydrostatics, control volume principles, Navier-Stokes equations, and flow kinematics (streamlines, vorticity).
• Two major flow classifications: inviscid flows (pressure and gravity forces dominate) vs viscous flows (friction forces matter); analysis methods differ fundamentally between these.
• Common confusion: potential flow vs ideal flow vs incompressible flow—potential flows are inviscid, incompressible, and irrotational (zero vorticity); ideal flows are inviscid and incompressible but not necessarily irrotational.
• Why it matters: combines mathematical rigor with physical interpretation to understand how forces and motion interact in fluids, applicable to transport, aerodynamics, and object motion through fluids.

🎯 Course positioning and approach

🎯 What this course is (and isn't)

• Second course: not introductory; builds on prior exposure to fundamental fluid concepts.
• Breadth over depth: exposes students to additional analysis techniques rather than exhaustive study of any single topic.
• Applications focus: aerodynamics (forces on arbitrary bodies), exact Navier-Stokes solutions, turbulent boundary layers and drag prediction.

🧮 Mathematical tools + physical insight

• Many mathematical tools will be developed and applied to specific flow situations.
• Early development of fluid mechanics came from mathematicians (Bernoulli, Euler, Navier, Stokes).
• Balance emphasized: the excerpt stresses learning not just equations and manipulation, but also physical situations and how flow phenomena are interpreted.
• Example: students should understand both the mathematical constraints and the physical consequences of a problem.

📚 Book organization preview

The excerpt outlines the structure:

• Chapter 2: mathematical tools (tensor notation, manipulations).
• Chapter 3: generalized Bernoulli equation with less restrictive conditions.
• Chapters 4–6: potential flow methods, solutions, and Panel Method for external flow forces.
• Chapters 7–10: viscous flows—Navier-Stokes equation, exact solutions, boundary layer equations (Blasius solution, integral methods), turbulence physics and scaling.

🌊 Fluid mechanics as a branch of mechanics

🌊 What mechanics studies

Mechanics: a branch of science dealing with forces and motion, and their relationships.

• Has static and dynamic elements: forces may exist without motion, with motion, or may initiate/change motion.
• Fluid response to forces is more complex than solids due to deformation rates over time.

🔄 Why fluids are complex

• Fluids yield to forces over time, creating complex distributions of pressure, velocity, and acceleration.
• Spatial distribution of fluid motion is key to understanding force transmission within fluids.
• Kinematic motion (how fluid moves) is a major area of study.
• Dynamic flow governed by Newton's momentum conservation: forces accompany rate of change of momentum.

🏛️ Historical context

• Origins date to ancient Greeks; Archimedes developed hydrostatics (forces by fluids, pressure changes due to gravity).
• Conservation of momentum formulations enabled analysis of fluid transport (pipelines, biological systems, lubrication, chemical reactions) and object motion within fluids.
• Newton's Law of reaction: forces on objects by fluids relate to forces by objects on fluids—"pushing fluid out of the way" creates mutual forces.

🔀 Major flow classifications

🔀 Inviscid vs viscous flows

The largest classification divides flows by whether friction matters:

Classification	Forces considered	Characteristics
Inviscid	Pressure distributions, gravity	Friction ignored; simpler analysis
Viscous	Pressure, gravity, and friction	Friction model required; more complex dynamics

• Inviscid approach: analyze forces on/by fluids and how they affect motion; pressure field influenced by fluid motion (pressure and velocity intertwined).
• Viscous approach: very different methods due to physical conditions, boundary conditions, and coupling of viscous effects with pressure and velocity.

🌀 Potential flows (a subset of inviscid)

Potential flows: irrotational flows that allow determination of flow and pressure fields using a scalar velocity potential.

• Key properties: inviscid, incompressible, and irrotational (vorticity = 0 throughout, except possibly at singularity points).
• Mathematical advantage: velocity vector replaced by scalar potential; viscous terms vanish; acceleration expression simplified.
• Trade-off: ignores frictional effects, so lost information about viscous forces that might alter flow/pressure; assumes these effects are minor for certain flow types.

💧 Ideal flows vs incompressible flows

• Ideal flows: inviscid and incompressible.
• Incompressible flows: no significant density changes with pressure changes.
  • Liquids: typically incompressible except under extreme high pressure.
  • Gases: may or may not be compressible depending on pressure changes within the flow (linked to velocity changes).
  • Don't confuse: density variations from temperature or solute concentration can occur in incompressible flows; "incompressible" refers only to pressure-density coupling.

🧩 How to distinguish flow types

• Potential flow = inviscid + incompressible + irrotational (requires velocity potential definition).
• Ideal flow = inviscid + incompressible (but not necessarily irrotational).
• Incompressible flow = density not significantly affected by pressure (but may be viscous or inviscid).

🧪 Viscous flow modeling

🧪 The viscosity model

• Viscous force model is not a universal law (unlike mass/energy conservation); it is mostly empirical, valid for a wide range of fluids and conditions but with exceptions for "exotic" fluids.
• Based on work by Navier and Stokes (1900s): formulation accounts for viscous effects in the Cauchy equation.
• Viscosity as a property: defines the viscous force needed for a specified fluid deformation rate.
• The excerpt restricts applications to incompressible, constant-property flows to assess term contributions; does not cover gas dynamics (compressible flows with equations of state).

🛑 No-slip boundary condition

No-slip boundary condition: at a fluid-solid interface, fluid velocity equals surface velocity in both magnitude and direction.

• Why it matters: moving away from the boundary, fluid velocity changes; the rate of change relates to frictional force on the surface by the fluid (and vice versa by Newton's third law).
• Continuum assumption: fluid treated at a scale larger than the mean free path of molecules (average distance before collision).
  • At the continuum scale, fluid velocity at the surface takes the surface velocity value.
  • Don't confuse: at sub-micron scales or in rarefied gases (very low pressure), continuum breaks down and "slip" may occur; kinetic theory of gases applies instead (beyond this course scope).

🔬 When the model breaks down

• Very small scale flows (sub-micron): continuum invalid, slip may occur.
• Rarefied gases (low pressure): molecules far apart, continuum not valid.
• Example: sticking your hand out of a car window vs out of a boat into water at the same speed—water density ~1000× air density, so resultant pressure ~1000× greater.

🧬 Fluid properties and their role

🧬 Why properties matter

• Numerical results depend heavily on density, viscosity, surface tension, compressibility.
• Density: important in pressure-velocity relationship.
  • Example: water density ~1000× air density → ~1000× greater pressure at same speed.
• Viscosity: highly viscous fluid (honey) vs low viscosity (water) have different flow characteristics when poured under gravity.

🧬 Scope of this course

• Fluid properties retained in problems to make distinctions obvious.
• Does not cover details of highly variable properties or resultant changes in flow/forces from variability.

📖 Content structure and learning outcomes

📖 What students will learn

After completion, students should better understand:

• How to analyze potential flows for incompressible conditions.
• How to analyze viscous flows for incompressible conditions.
• How forces and motion interact in fluids, with applications to transport, aerodynamics, and object motion.

📖 Chapter roadmap

1. Chapter 2: Mathematical tools (tensor notation, manipulations).
2. Chapter 3: Generalized Bernoulli equation (less restrictive conditions than first-course version).
3. Chapters 4–5: Potential flow methods and solutions.
4. Chapter 6: Panel Method for pressure and forces on objects in external flow.
5. Chapter 7: Navier-Stokes equation development.
6. Chapter 8: Classical "exact" solutions; boundary layer equations; Blasius solution.
7. Chapter 9: Integral solution method.
8. Chapter 10: Turbulence—basic physics, scaling conditions, application to boundary layer flows.

⚠️ Note on text status

• This text is archived and will not be updated; content may be outdated.
• Compiled from notes for a course offered to seniors and first-year graduate students with mechanical engineering or related background.

🧭 Overview

🧠 One-sentence thesis

Tensor notation provides a powerful, compact way to express fluid mechanics equations by representing vectors and higher-order quantities through indexed components, enabling clearer manipulation of gradient, divergence, and acceleration terms that describe how fluids move and deform.

📌 Key points (3–5)

• Tensor notation basics: scalars are zero-order tensors, vectors are first-order tensors (3 components), and stress is a second-order tensor (9 components); subscripts indicate which component and coordinate direction.
• Key operators: gradient produces vectors from scalars (or tensors from vectors), divergence produces scalars from vectors, and the material derivative tracks how properties change for a moving fluid mass.
• Velocity gradient decomposition: the velocity gradient tensor splits into a symmetric part (strain rate/deformation) and an antisymmetric part (rotation rate, related to vorticity).
• Common confusion: repeated indices mean summation over all coordinate values; free indices indicate the result is still a vector or tensor with that index.
• Streamline coordinates: acceleration along and normal to streamlines includes convective terms and curvature effects, with normal acceleration proportional to velocity squared divided by radius of curvature.

📐 Tensor notation fundamentals

📐 What tensors represent

A scalar is a zero-order tensor, a vector is a first-order tensor, and stress is a second-order tensor.

• Order determines components:
  • Zero-order (scalar): 1 component (just a magnitude).
  • First-order (vector): 3 components in 3D space.
  • Second-order: 9 components (3×3 combinations).
  • Third-order: 27 components (3×3×3 combinations).
• Subscripts indicate which component: for velocity V, subscript i can be 1, 2, or 3, representing x, y, z (or r, theta, z in cylindrical).
• Example: velocity vector V in tensor notation is V_i, where i=1 gives the x-component, i=2 gives y, i=3 gives z.

🔢 Index rules

• Repeated index = summation: when the same subscript appears twice in one term, sum over all three values.
  • Example: ∂u_i/∂x_i means ∂u_1/∂x_1 + ∂u_2/∂x_2 + ∂u_3/∂x_3 (divergence of velocity).
• Free index = vector/tensor: an index that appears once indicates the result is still a vector or tensor.
  • Example: a_i b_i has repeated i (scalar result); a_i b_j has free i and j (second-order tensor result).
• Don't confuse: subscripts are not unit vectors; they label which component or direction.

🔧 Special operators

Kronecker delta δ_ij:

• δ_ij = 1 when i = j, otherwise δ_ij = 0.
• Used to select only matching components.
• Example: including pressure gradient as δ_ij ∂p/∂x_j picks out only the i-direction component.

Permutation operator ε_ijk:

• ε_ijk = +1 if indices are in cyclic order (1,2,3 or 2,3,1 or 3,1,2).
• ε_ijk = -1 if indices are in anti-cyclic order (1,3,2 or 2,1,3 or 3,2,1).
• ε_ijk = 0 if any two indices are the same.
• Used for cross products and curl operations.

🧮 Key differential operators

🧮 Gradient operator

The gradient is a vector operator that takes partial derivatives with respect to each coordinate component.

• In Cartesian coordinates: gradient = (∂/∂x, ∂/∂y, ∂/∂z).
• Tensor notation: ∂/∂x_i.
• Gradient of a scalar produces a vector (3 components).
  • Example: gradient of pressure p gives a vector pointing in the direction of steepest pressure increase.
• Gradient of a vector produces a second-order tensor (9 components).
  • Tensor notation: ∂u_j/∂x_i (i is the derivative direction, j is the velocity component).

🌊 Divergence operator

Divergence is the dot product of the gradient operator with a vector, resulting in a scalar.

• Vector notation: divergence of V = ∇·V.
• Tensor notation: ∂u_i/∂x_i (repeated index i means sum).
• In Cartesian: ∂u/∂x + ∂v/∂y + ∂w/∂z.
• Divergence of a second-order tensor yields a vector.
  • Example: x-component is ∂u_1/∂x_1 + ∂u_2/∂x_2 + ∂u_3/∂x_3; y and z components replace u_1 with u_2 and u_3.

🌀 Curl operator

• Curl is the cross product of the gradient operator with a vector, resulting in a vector (pseudo-vector).
• Tensor notation uses the permutation operator: ε_ijk ∂u_k/∂x_j.
• Curl measures rotation rate in the flow.

⏱️ Material derivative

⏱️ What it tracks

The material derivative expresses the rate of change of a property (scalar or tensor) for a fixed mass of fluid as it moves through space and time.

• Notation: D/Dt (capital D over capital T).
• Applies to any property: pressure, temperature, velocity, etc.
• Based on a Lagrangian frame: you follow a specific fluid mass.

⏱️ The formula

For a scalar property p (pressure):

• Dp/Dt = ∂p/∂t + u ∂p/∂x + v ∂p/∂y + w ∂p/∂z.
• Tensor notation: Dp/Dt = ∂p/∂t + u_i ∂p/∂x_i.
• First term (∂p/∂t): local rate of change at a fixed point in space.
• Remaining terms (u_i ∂p/∂x_i): convective rate of change due to the fluid moving through a spatially varying field.

For a vector (velocity V):

• DV/Dt = ∂V/∂t + (V·∇)V.
• Tensor notation: Du_i/Dt = ∂u_i/∂t + u_j ∂u_i/∂x_j.
• Each component of velocity has its own material derivative.

⏱️ Cylindrical coordinates

• In cylindrical coordinates (r, θ, z), the unit vectors e_r and e_θ change direction as θ changes.
• Material derivative includes extra terms to account for changing unit vectors.
• Result: DV/Dt = ∂V/∂t + u_r ∂V/∂r + (u_θ/r) ∂V/∂θ + u_z ∂V/∂z + (terms for unit vector changes).
• The unit vector changes contribute terms like u_θ²/r (centripetal acceleration).

🚀 Acceleration

🚀 Acceleration as material derivative of velocity

Acceleration of a fluid mass is the material derivative of the velocity vector.

• Tensor notation: Du_i/Dt = ∂u_i/∂t + u_j ∂u_i/∂x_j.
• Local acceleration (∂u_i/∂t): velocity change at a fixed point in space.
  • Zero for steady flow (by definition).
• Convective acceleration (u_j ∂u_i/∂x_j): velocity change due to moving through a spatially varying velocity field.
  • Example: steady flow through a nozzle has zero local acceleration but nonzero convective acceleration (fluid speeds up as it moves).

🚀 Rewriting convective acceleration

• Vector identity: (V·∇)V = ∇(V²/2) - V×(∇×V).
• Tensor notation: u_j ∂u_i/∂x_j = ∂(u_k u_k/2)/∂x_i - ε_ijk u_j ω_k.
• This splits convective acceleration into a gradient of kinetic energy and a term involving vorticity (ω = curl of V).
• Advantage: helps physical interpretation and simplifies some equations.

🚀 Streamline coordinates

Streamline coordinates define directions along (s) and normal (n) to the velocity vector.

• Along streamline (s direction):
  • a_s = ∂V_s/∂t + V_s ∂V_s/∂s.
  • Local + convective acceleration along the flow direction.
• Normal to streamline (n direction):
  • a_n = V_s²/R.
  • R is the local radius of curvature of the streamline.
  • Direction is inward toward the center of curvature.
  • For a straight streamline, R → infinity, so a_n → 0.
• Don't confuse: velocity in the n direction (V_n) is zero by definition of a streamline, but acceleration in the n direction is not zero when the streamline curves.

🔀 Velocity gradient tensor

🔀 What it represents

The velocity gradient tensor ∂u_j/∂x_i is a second-order tensor (9 components) describing how velocity varies near a point in the flow.

• It is a local, point function (may also be time-dependent).
• Tensor notation: ∂u_j/∂x_i (i is the spatial direction of the derivative, j is the velocity component).

🔀 Decomposition into symmetric and antisymmetric parts

Any second-order tensor can be split:

• Symmetric part: (1/2)(∂u_j/∂x_i + ∂u_i/∂x_j).
  • Represents strain rate or deformation rate of the fluid element.
• Antisymmetric part: (1/2)(∂u_j/∂x_i - ∂u_i/∂x_j).
  • Represents rotation rate of the fluid element.
  • Related to vorticity: antisymmetric part = (1/2) × curl of V.

🔀 Rotation and vorticity

• The antisymmetric part is a second-order tensor, but diagonal elements (i=j) are zero.
• Off-diagonal elements with swapped indices are negatives of each other.
• Result: only 3 independent values → a pseudo-vector (vorticity).
• Vorticity component: ω_k = ε_ijk ∂u_j/∂x_i = curl of V.
• Sign convention: counterclockwise rotation is positive.
• Example: if ∂u_2/∂x_1 is positive (velocity in y-direction increases as you move in x-direction), the flow tends to rotate clockwise (negative vorticity about the z-axis).

🔀 Strain rate

• When the two velocity gradient components contributing to a rotation have equal magnitude and the same sign, the strain rate is zero and the flow is in pure rotation (like a solid body).
• When rotation rates of vertical and horizontal axes differ, there is net deformation (strain).

📊 Summary table: Vector vs tensor notation

Quantity	Vector Notation	Tensor Notation	Order
Scalar	a	a	0
Vector	V	u_i	1
Gradient of scalar	∇a	∂a/∂x_i	1 (vector)
Gradient of vector	∇V	∂u_j/∂x_i	2 (tensor)
Divergence of vector	∇·V	∂u_i/∂x_i	0 (scalar)
Curl of vector	∇×V	ε_ijk ∂u_k/∂x_j	1 (pseudo-vector)
Material derivative	D/Dt	D/Dt	operator
Dot product	A·B	a_i b_i	0 (scalar)
Cross product	A×B	ε_ijk a_j b_k	1 (vector)

🧪 Physical interpretation

🧪 Material derivative in practice

• Steady flow through a nozzle: ∂V/∂t = 0 (steady), but DV/Dt ≠ 0 because the fluid accelerates as it moves through the converging area (convective acceleration).
• Unsteady flow at a fixed point: ∂V/∂t ≠ 0 (local acceleration), even if the spatial gradient is zero.

🧪 Streamline curvature

• When a streamline curves, the fluid experiences centripetal acceleration V²/R directed inward.
• Example: flow around a bend—velocity may be constant in magnitude along the streamline, but direction changes, so there is acceleration normal to the streamline.
• Don't confuse: straight streamline (R = infinity) has zero normal acceleration; curved streamline always has nonzero normal acceleration if velocity is nonzero.

🧪 Vorticity and rotation

• Pure rotation: velocity gradient components cause equal rotation rates in both axes → strain rate is zero, vorticity is nonzero.
• Pure strain: velocity gradient components cause deformation without net rotation → vorticity is zero, strain rate is nonzero.
• Example: a square fluid element can rotate (all sides rotate at the same rate) or deform (opposite sides stretch/compress without rotating).

🧭 Overview

🧠 One-sentence thesis

The Bernoulli equation relates acceleration, pressure, and gravitational forces in a fluid, and its generalized form can be applied to unsteady, compressible, and rotational flows when viscous forces are negligible, while the simplified form requires incompressible, steady flow along a streamline.

📌 Key points (3–5)

• What the Bernoulli equation describes: the balance between velocity (kinetic energy), pressure, and elevation (potential energy) in a fluid flow, derived from the momentum equation without viscous forces.
• Generalized vs simplified forms: the generalized form (Equation 3.10) applies to unsteady, compressible flow in any direction; the simplified form (Equation 3.1) requires incompressible, steady flow along a streamline.
• Role of vorticity and irrotational flow: when vorticity is zero everywhere (irrotational flow), a velocity potential exists and the Lamb vector term vanishes, simplifying the equation for any integration path, not just streamlines.
• Common confusion—streamline vs irrotational: the simplified Bernoulli equation applies along a streamline (even if the flow is rotational elsewhere) or anywhere in an irrotational flow; these are two different ways to eliminate the Lamb vector term.
• Key restriction across all forms: viscous (frictional) forces are always excluded; inviscid flow is assumed, meaning the equation cannot be applied where fluid friction is significant.

📐 The generalized Bernoulli equation

📐 Starting point: Euler's equation

Euler's equation for fluid motion: the differential equation relating fluid acceleration to pressure forces and gravitational body forces per unit mass, excluding viscous effects.

• The material derivative (total acceleration) on the left balances pressure gradient and gravity on the right.
• Tensor notation form: the pressure term represents net compressive force per unit mass along each vector component direction.
• The body force term is the gravitational vector component in the direction of interest.

🔄 Introducing vorticity and the Lamb vector

• The convective acceleration term is rewritten using a vector identity: it splits into the gradient of half the velocity magnitude squared plus the cross product of velocity and vorticity.
• Vorticity is defined as the curl of the velocity vector; it measures local rotation in the flow.
• Lamb vector: the cross product of velocity and vorticity, named after mathematician Horace Lamb (1849–1934).
• This reformulation allows Euler's equation to be expressed in terms of gradients and the Lamb vector, making integration along any path possible.

🧮 Gravitational potential and spatial integration

• The gravitational term is rewritten as a gradient of a scalar potential, where the potential has units of length.
• If gravity is vertically downward, the potential represents elevation above an arbitrary datum (upward direction).
• Example: if the integration path is horizontal, the change in elevation is zero; if vertical upward, the change equals the path length; if at an angle, the change is proportional to the sine of the angle.
• Each term in Euler's equation is then integrated along an arbitrary spatial path (elemental distance), taking the dot product of each term with the path direction vector.

📝 The generalized form (Equation 3.10)

After spatial integration along an arbitrary line, the generalized Bernoulli equation is:

• Left side: local acceleration integral, kinetic energy term (half velocity squared), pressure integral, and gravitational potential change.
• Right side: a general function of time only (vanishes if flow is steady).
• Only restriction: viscous forces are excluded; no other assumptions about compressibility, steadiness, or flow direction are made.
• This form applies to unsteady, compressible flow along any chosen path, as long as friction is negligible.

🔧 Simplifications and special cases

🔧 Incompressible flow

• If density is constant, the pressure integral simplifies: the integral of pressure gradient over density becomes simply the pressure change divided by constant density.
• The generalized equation becomes easier to evaluate because the pressure term no longer depends on how density varies along the path.

⏱️ Steady flow

• If there is no time dependence, the local acceleration term (first integral on the left) and the time function on the right both vanish.
• The equation reduces to a balance of kinetic energy, pressure, and gravitational potential changes along the path.

🌀 Along a streamline or irrotational flow

• Along a streamline: the velocity vector is tangent to the streamline by definition, so the integration direction and velocity are aligned; the Lamb vector (cross product of velocity and vorticity) is perpendicular to the streamline, so its projection along the streamline is zero.
• Irrotational flow: if vorticity is zero everywhere, the Lamb vector vanishes for any integration path, not just streamlines.
• Don't confuse: applying the equation along a streamline works even if the flow is rotational elsewhere; irrotational flow allows application between any two points in the field.

🧩 Velocity potential in irrotational flow

Velocity potential: a scalar function whose spatial partial derivatives define the velocity components in each direction.

• If a velocity potential exists, taking its derivatives and computing vorticity yields zero identically (because the order of differentiation can be reversed).
• Conversely, if vorticity is zero everywhere, a velocity potential can be defined to describe the entire velocity field.
• Irrotational flow means vorticity-free flow, equivalent to zero local rotation; the flow can be fully described by a single scalar potential.
• Using the velocity potential, the unsteady Bernoulli equation can be rewritten with the time derivative of the potential replacing the velocity in the local acceleration term.

📋 Summary of Bernoulli equation forms

Form	Conditions	Key features
General (3.10)	Inviscid only	Unsteady, compressible, any path; includes local acceleration integral, pressure integral, Lamb vector term
Incompressible	Inviscid + constant density	Pressure term simplifies to pressure change over constant density
Steady	Inviscid + no time dependence	Local acceleration and time function vanish
Irrotational or streamline	Inviscid + (zero vorticity everywhere or integration along streamline)	Lamb vector term vanishes; applies between any two points (if irrotational) or along a streamline (if rotational)
Simplified (3.1)	Inviscid + incompressible + steady + along streamline (or irrotational)	Sum of velocity squared over two, pressure over density, and elevation is constant; most commonly taught form

🚨 Common restrictions and confusions

• All forms exclude viscous forces: the Bernoulli equation cannot account for internal fluid friction; it assumes inviscid flow.
• Inviscid ≠ irrotational: inviscid flow (no friction) does not automatically mean irrotational flow (zero vorticity); however, irrotational flow does imply inviscid flow.
• Boundary conditions in inviscid flow: because viscous forces are excluded, the no-slip boundary condition (fluid velocity equals surface velocity) does not apply; inviscid flows allow slip at boundaries, and surface velocity may need to be determined separately.
• Streamline application: the simplified form is often presented as requiring flow "along a streamline," but this is only necessary if the flow is rotational; if the flow is irrotational, the equation applies between any two points.

🔬 Unsteady incompressible form with velocity potential

🔬 Replacing velocity with potential

• For unsteady, incompressible, irrotational flow, the velocity in the local acceleration term is replaced by the gradient of the velocity potential.
• The order of spatial integration and time derivative is interchanged, bringing the time derivative inside the spatial difference.
• The result is an unsteady Bernoulli equation expressed entirely in terms of the velocity potential and its time derivative, pressure, and gravitational potential.

🔬 Why the Lamb vector vanishes

• Introducing the velocity potential automatically enforces irrotational flow, because the curl of a gradient (velocity potential) is zero by vector identity.
• Therefore, the Lamb vector term is eliminated without needing to specify integration along a streamline.

🎯 Practical implications

🎯 When to use which form

• Known upstream/downstream conditions: apply the Bernoulli equation between a point where flow properties are known (e.g., far upstream) and a point where properties are desired (e.g., on an object surface).
• Rotational flow: must apply along a streamline; identify the streamline connecting the two points.
• Irrotational flow: can apply between any two points in the field; no need to trace a streamline.
• Unsteady problems: use the generalized or unsteady incompressible form; include the local acceleration or velocity potential time derivative term.

🎯 Limitations and care in application

• The simplified form (Equation 3.1) is powerful but highly restrictive: incompressible, steady, along a streamline (or irrotational), and inviscid.
• Example: applying to a liquid (nearly incompressible) or low-velocity gas (small pressure changes) is valid; applying to high-speed gas flow with large pressure changes requires the compressible form.
• Viscous effects: if spatial velocity gradients are large or viscosity is significant, the Bernoulli equation is not valid; pressure and body forces must dominate over friction.
• Slip boundary condition: in inviscid flow analysis, do not enforce no-slip; the fluid velocity at a surface is not necessarily equal to the surface velocity and may need to be determined as part of the solution.

🧭 Overview

🧠 One-sentence thesis

Potential flow theory uses scalar functions—the velocity potential and streamfunction—to describe irrotational, incompressible flows, both of which satisfy the Laplace equation and provide orthogonal representations of the velocity field.

📌 Key points (3–5)

• What potential flow is: flow situations where vorticity is zero throughout (irrotational), allowing a scalar velocity potential φ to describe the entire velocity field.
• Continuity equation: conservation of mass applied to a fluid element; for incompressible flow it reduces to the divergence of velocity being zero.
• Streamfunction ψ: a scalar that defines streamlines; flow rate between two streamlines equals the change in ψ, and it automatically satisfies continuity for incompressible flow.
• Common confusion: velocity potential vs streamfunction—velocity is tangent to constant-ψ lines (streamlines) but normal to constant-φ lines; the two families of lines are orthogonal.
• Why it matters: both φ and ψ satisfy Laplace's equation for ideal flow, enabling mathematical solutions and flow visualization through streamline spacing (closer streamlines = higher velocity).

🌊 What is potential flow

🌊 Definition and core idea

Potential flows are those flow situations where the flow is taken to be irrotational, such that the vorticity is zero throughout the flow field (except at possible singularity points).

• Irrotational means the fluid elements do not rotate locally; vorticity = 0.
• This allows the use of a scalar function φ (velocity potential) to describe the entire flow field.
• The velocity potential is defined by: each component of velocity equals the local spatial partial derivative of φ in the direction of that velocity component.
• Example: the x-component of velocity is the partial derivative of φ with respect to x.

🔄 Why irrotational flow allows a potential

• By a vector identity, inserting the velocity potential definition into the definition of vorticity results in vorticity being identically zero.
• This is stated in chapter 2 of the source material.
• The excerpt notes that the term involving vorticity is eliminated when introducing the velocity potential because irrotational flow is required.

📐 Continuity equation and conservation of mass

📐 Deriving continuity from mass conservation

• The continuity equation comes from conservation of mass applied to a continuum of fluid that may be in motion.
• Basic physics: no mass can be created or destroyed over time.
• Net flow in minus net flow out must equal the net change in mass within a control volume.

🧊 Cubic element derivation

• Imagine a three-dimensional cubic volume in space.
• Each face can have mass flow across it; mass flow rate across any area = density × velocity normal to the area × area.
• For each direction (x, y, z), calculate outflow minus inflow using the change of (density × velocity) between faces.
• Summing all three directions and setting equal to the change of mass within the volume gives the continuity equation.

🧮 General and incompressible forms

General continuity equation:

• Time derivative of density plus spatial derivatives of (density × velocity) in all three directions = 0.
• If flow is steady, the time derivative term is zero.
• If density is constant (incompressible), density factors out and the time derivative of density is zero.

Incompressible continuity:

• Reduces to: sum of spatial derivatives of velocity components = 0 (divergence of velocity = 0).
• This form does not require steady flow; velocity may vary with time.
• Don't confuse: incompressible continuity has no time derivative of density, but velocity can still be unsteady.

🔗 Continuity with velocity potential

• Insert the velocity potential definition (velocity = derivative of φ) into the incompressible continuity equation.
• Result: the Laplace operation on φ equals zero (sum of second partial derivatives of φ in all directions = 0).
• This represents continuity (conservation of mass) for incompressible flow using the velocity potential.

🌀 Streamfunction and streamlines

🌀 What is a streamfunction

The streamfunction ψ is a scalar quantity associated with each streamline; the equation of a streamline can be represented by a constant value of ψ.

• For simplicity, introduced in two dimensions but valid in three dimensions.
• A streamline is a line tangent to the velocity vector at every point (instantaneous for time-dependent flow).
• The streamfunction value is constant along a streamline.

📏 Definition in terms of velocity

• For a constant ψ along a streamline, small changes satisfy: (change in ψ) = 0.
• The slope of the streamline equals the ratio of y-velocity to x-velocity (since velocity is tangent to the streamline).
• This leads to the definition: partial derivative of ψ with respect to y = x-velocity component; negative partial derivative of ψ with respect to x = y-velocity component.
• If you know velocity components as functions of position, integrate to find ψ; if you know ψ, differentiate to find velocity components.

✅ Streamfunction automatically satisfies continuity

• Insert the streamfunction definition into the incompressible continuity equation.
• Result: an identity (automatically true).
• Interpretation: if the streamfunction exists by its definition, the flow automatically satisfies continuity for incompressible conditions.

🎯 Stagnation points and streamline intersections

• At any given point there is a velocity vector and therefore a streamline passing through it.
• Two or more streamlines can intersect at a point only if the velocity magnitude is zero (both partial derivatives of ψ are zero, slope undefined).
• A stagnation point is such an intersection; the streamfunction can be continuous through it, but flow divides (one branch up, another down).
• Stagnation points can be on surfaces or distributed within the flow field.

💧 Flow rate and streamfunction interpretation

💧 General flow rate through an area

• Mass flow rate through area A with outward normal vector n̂ = density × (velocity · n̂) × A.
• The dot product (velocity · n̂) is the projection of velocity in the outward normal direction.
• Volume flow rate Q = mass flow rate / density.

🔀 Flow rate between streamlines

• In two-dimensional steady flow, velocity is tangent to each streamline, so there is no flow across a streamline.
• Flow between two streamlines must remain between those streamlines along the flow direction.
• The flow rate between two streamlines remains constant.

📊 Change in streamfunction equals flow rate

• Consider two streamlines with streamfunction values ψ₁ and ψ₂.
• Draw a control volume with flow entering through two areas and exiting through another.
• Volumetric flow rate per unit depth in must balance flow rate out.
• Result: flow rate between two points = change in streamfunction value between those points (ψ₂ − ψ₁).
• This provides both a mathematical and physical interpretation for the streamfunction.

🔍 Visualizing velocity from streamline spacing

• In a wind tunnel test, inject smoke at discrete upstream points separated vertically; smoke lines trace streamlines.
• As streamlines converge (distance between them decreases), area decreases, so velocity must increase.
• As streamlines diverge, velocity decreases.
• The relationship between cross-sectional area and velocity is linear.
• Measuring the change of distance between adjacent streamlines provides a measure of velocity change.
• Example: smoke flow visualization over an inclined flat wing shows streamline convergence and divergence.

🔗 Vorticity, Laplace equation, and orthogonality

🔗 Vorticity and streamfunction

• Insert the streamfunction definition into the definition of vorticity for two-dimensional flow in the x-y plane.
• Result: vorticity = negative of the Laplace of the streamfunction (sum of second partial derivatives of ψ).
• For irrotational flow (vorticity = 0), the Laplace of ψ = 0.

⚖️ Both φ and ψ satisfy Laplace's equation

• For irrotational, incompressible (ideal) flow:
  • Laplace of velocity potential φ = 0.
  • Laplace of streamfunction ψ = 0.
• Both have the same functional form (Laplace equation).
• This enables solving the Laplace equation for either quantity and then determining the velocity field from the definitions of φ and ψ.

⊥ Orthogonality of φ and ψ lines

• Velocity is tangent to constant-ψ lines (streamlines).
• Velocity is normal to constant-φ lines (velocity potential lines).
• Consequently, lines of constant ψ are orthogonal (perpendicular) to lines of constant φ.
• The slope of constant-φ lines is (negative y-velocity / x-velocity); the slope of constant-ψ lines is (y-velocity / x-velocity).
• Don't confuse: both describe the same flow, but from orthogonal perspectives—one shows flow direction (streamlines), the other shows equipotential contours.

🧭 Cylindrical coordinate expressions

🧭 Two-dimensional incompressible flow in cylindrical coordinates

The excerpt provides a table of expressions in cylindrical coordinates (r = radial, θ = circumferential):

Quantity	Expression
Continuity	Divergence of velocity in r and θ directions = 0
Streamfunction	Radial velocity = (1/r) × (partial derivative of ψ with respect to θ); circumferential velocity = negative partial derivative of ψ with respect to r
Vorticity	Only z-component exists for 2D flow in r-θ plane; all other components are zero
Velocity Potential	Radial velocity = partial derivative of φ with respect to r; circumferential velocity = (1/r) × (partial derivative of φ with respect to θ)

• These expressions are the cylindrical-coordinate analogs of the Cartesian definitions.
• They allow analysis of flows with circular or radial symmetry.

🧭 Overview

🧠 One-sentence thesis

Potential flows can be constructed by superimposing basic ideal flow elements (uniform flow, sources, sinks, vortices, doublets) because the governing Laplace equations are linear, enabling the modeling of complex flows around objects like cylinders and airfoils.

📌 Key points (3–5)

• Superposition principle: Basic ideal flows can be linearly added to create more complex flows because the underlying equations (Laplace equations for streamfunction and velocity potential) are linear.
• Orthogonality of descriptors: Streamfunction lines and velocity potential lines are orthogonal; velocity is tangent to streamfunction lines and normal to potential lines.
• Basic flow elements: Uniform flow, source/sink, vortex, and doublet are fundamental building blocks; each satisfies the Laplace equation and has specific streamfunction and velocity potential forms.
• Common confusion: A free vortex appears to have circulation (nonzero) yet is irrotational (zero vorticity everywhere except at the center singularity)—vorticity is concentrated at r = 0, so the flow away from the center is irrotational.
• Practical application: Combining these elements (e.g., uniform flow + doublet = flow over a cylinder; adding circulation generates lift via the Kutta-Joukowsky law).

🔗 Streamfunction and velocity potential relationship

🔗 Orthogonality condition

Lines of constant streamfunction (ψ) are orthogonal to lines of constant velocity potential (φ).

• Both ψ and φ satisfy the Laplace equation for ideal (irrotational, incompressible, inviscid) flows.
• The slope of constant potential lines is u/v, while the slope of constant streamfunction lines is −v/u.
• Velocity is always along the streamfunction line and normal to the potential lines.
• Why it matters: Knowing one (ψ or φ) makes it straightforward to determine the other; the excerpt uses mostly streamfunction representation.

🧮 Governing equations

• For ideal flows, the simplified continuity equation treats density as constant, leading to conditions on velocity components to satisfy mass conservation.
• Irrotational flow requires vorticity to be zero, imposing additional conditions on velocity derivatives.
• Boundary conditions: inviscid flow does not have no-slip; instead, surfaces are impermeable (normal velocity component = 0 at the surface).

🧩 Basic flow elements

🧩 Uniform flow

Uniform flow: a steady flow with constant velocity in a given direction, so the velocity vector does not vary spatially.

• Designate velocity as U; align coordinates so U is in the x direction.
• Streamfunction: ψ = U y (setting ψ = 0 at y = 0).
• Velocity potential: φ = U x (setting φ = 0 at x = 0).
• For uniform flow at angle α to the x axis: ψ = U (y cos α − x sin α).
• Volumetric flow rate per depth between y₁ and y₂: Q = U (y₂ − y₁).

🌀 Source/sink flow

Source/sink flow: radial flow from (source) or into (sink) a point; mass flow rate per unit depth is m, with source strength Λ = m / (2π).

• Source has positive strength (outward radial velocity); sink has negative strength (inward).
• Radial velocity: v_r = Λ / r (velocity decreases linearly as r increases; infinite at r = 0).
• Streamfunction: ψ = Λ θ (where θ is the angle in cylindrical coordinates).
• Velocity potential: φ = Λ ln r.
• Streamlines are straight radial lines; potential lines are circles.
• Don't confuse: The singularity at r = 0 (infinite velocity, zero area) is unrealistic, but moving away from it yields a well-defined flow rate.

🌪️ Vortex flow (free vortex)

Free vortex: swirling flow with circular streamlines, no radial velocity component, and irrotational everywhere except at the center.

• Circumferential velocity: v_θ = C / r (where C is a constant).
• Circulation Γ (line integral of velocity around a closed loop): Γ = 2π C; vortex strength K = Γ / (2π).
• Streamfunction: ψ = −K ln r (or ψ = −(Γ / 2π) ln r).
• Velocity potential: φ = K θ (or φ = (Γ / 2π) θ).
• Streamlines are circles; potential lines are straight radial lines (inverse of source/sink).
• Sign convention: Counterclockwise rotation is positive circulation; clockwise is negative.

🧲 Vortex paradox: circulation vs vorticity

• A free vortex has finite circulation Γ but zero vorticity everywhere except at r = 0.
• How to reconcile: Circulation is the area integral of vorticity. For any two circular contours of different radii (both centered at the vortex), the circulation is the same, so no vorticity exists between them.
• Vorticity is concentrated at the center (a singularity); the flow away from the center is irrotational.
• Example: Draw arbitrarily small circles around the center—circulation remains constant, implying no vorticity in the region between circles.

🔗 Doublet

Doublet: the result of superimposing a source and sink of equal strength, brought very close together (separation distance a → 0) while keeping the product (source strength × separation) constant.

• Streamfunction: ψ = −(μ sin θ) / r, where μ is the doublet strength.
• Rearranging: r sin θ − (μ / ψ) = 0, which is the equation of a circle centered along the y axis.
• Each streamline (constant ψ) is a circle tangent to the x axis at the origin; radius = μ / ψ.
• Velocity potential lines are also circles (orthogonal to streamlines).

🏗️ Superposition and complex flows

🏗️ Why superposition works

• The Laplace equation (governing ψ and φ) is linear, so solutions can be added.
• Basic flows (uniform, source, sink, vortex, doublet) are solutions of the Laplace equation.
• Combining them (with appropriate strengths and positions) creates more complex flows that still satisfy the governing equations.

🏗️ Source + sink = flow field

• Place a source at (−a, 0) and a sink at (+a, 0), both on the x axis, with equal but opposite strength.
• Streamfunction at any point P: ψ = Λ (θ₂ − θ₁), where θ₁ and θ₂ are angles from the source and sink to P.
• Velocity potential: φ = Λ (ln r₂ − ln r₁).
• Streamlines are symmetric about the x and y axes.

🏗️ Rankine Oval

• Add uniform flow (velocity U in +x direction) to the source + sink combination.
• Streamfunction: ψ = U y + Λ (θ₂ − θ₁).
• Setting ψ = 0 yields an oval-shaped streamline (the Rankine Oval).
• Application: Models flow over an oval surface with approach velocity U; adjust source/sink strength and separation to change the oval geometry.

🏗️ Uniform flow + doublet = flow over a cylinder

• Streamfunction: ψ = U r sin θ − (μ sin θ) / r.
• Setting ψ = 0 at r = a (where a² = μ / U) defines a circle of radius a.
• This models uniform flow over a cylinder of radius a.
• Velocity components (cylindrical): v_r = U cos θ (1 − a² / r²); v_θ = −U sin θ (1 + a² / r²).
• At r = a, v_r = 0 (no flow through the surface); v_θ ≠ 0 (velocity varies around the cylinder).
• Stagnation points (v = 0): at θ = 0 and θ = π (front and back of the cylinder).
• Streamlines are symmetric about x and y axes, implying zero net force (no drag or lift without circulation).

🎯 Circulation and lift

🎯 Adding circulation to cylinder flow

• Superimpose uniform flow, doublet, and vortex: ψ = U r sin θ − (μ sin θ) / r − K ln r.
• Setting ψ = 0 at r = a (cylinder surface) and choosing K appropriately yields flow over a rotating cylinder.
• Circulation Γ = 2π K; for clockwise rotation, Γ is negative.
• Stagnation points move: solving v_r = 0 and v_θ = 0 gives sin θ_stag = −Γ / (4π U a).
• For Γ = 0, stagnation points are at θ = 0, π (symmetric). For Γ ≠ 0, stagnation points shift (asymmetric flow).

🎯 Kutta-Joukowsky Law

Kutta-Joukowsky Law: The lift force per unit depth on a two-dimensional body in inviscid flow is L = ρ U Γ, where ρ is density, U is freestream velocity, and Γ is circulation.

• Derived by integrating the pressure distribution around the body (using Bernoulli's equation).
• For a rotating cylinder: L = ρ U Γ (per unit depth into the page).
• Sign convention: Counterclockwise circulation (positive Γ) with flow left-to-right produces downward force; clockwise (negative Γ) produces upward force.
• Why it matters: Circulation creates asymmetry in the flow (higher velocity on one side, lower pressure by Bernoulli), resulting in a net lift force.
• Applies to other shapes (e.g., airfoils) as well, not just cylinders.
• Don't confuse: This is inviscid (no viscous drag); it only accounts for pressure forces and is accurate when flow does not separate from the surface.

🎯 Kutta condition for airfoils

• At the trailing edge of an airfoil, the Kutta condition requires that flow exits the top and bottom surfaces with equal velocity and pressure.
• This prevents flow from wrapping around the sharp trailing edge.
• The Kutta condition is the boundary condition that determines the circulation (and thus lift) for a given angle of attack.

🖼️ Imaging method

🖼️ Simulating solid boundaries

Imaging: A method to insert impervious walls by using a streamline as a solid boundary, since no flow crosses a streamline.

• Example: A source near a flat wall (the x axis).
• Place a mirror-image source of equal strength on the opposite side of the wall.
• The superposition creates symmetric flow; the wall (x axis) becomes a streamline (ψ = 0).
• Velocity component normal to the wall is zero; tangential component is nonzero.
• Application: Models pumping fluid from a well near a rock formation boundary (or flow into a well, using a sink).

🧮 Complex variable representation

🧮 Complex potential

Complex potential F(z): A function F = φ + i ψ, where φ is velocity potential, ψ is streamfunction, and z = x + i y (or z = r e^(i θ) in cylindrical coordinates).

• The real part of F is φ; the imaginary part is ψ.
• Both φ and ψ satisfy the Laplace equation and are harmonic functions.
• Cauchy-Riemann conditions: ∂φ/∂x = ∂ψ/∂y and ∂φ/∂y = −∂ψ/∂x (ensuring orthogonality).

🧮 Complex velocity

• The derivative dF/dz = u − i v, where u and v are velocity components in x and y directions.
• In cylindrical coordinates: dF/dz = (v_r − i v_θ) e^(−i θ).
• Magnitude of velocity: |V| = sqrt(u² + v²) = sqrt(dF/dz × conjugate(dF/dz)).
• Once velocity is known, pressure is found from the Bernoulli equation.

🧮 Examples in complex form

Flow type	Complex potential F(z)	Notes
Uniform flow (velocity U in +x)	F = U z	Real part φ = U x; imaginary part ψ = U y
Uniform flow at angle α	F = U e^(−i α) z	Rotates the flow by angle α
Source/sink (strength Λ)	F = (Λ / 2π) ln z	Source if Λ > 0; sink if Λ < 0
Vortex (strength K)	F = −i (K / 2π) ln z	Negative sign → counterclockwise rotation
Doublet (strength μ)	F = μ / z	Streamlines are circles tangent to the origin

🧮 Flow in a sector

• General form: F = A z^n, where n is a parameter and A is a constant.
• Streamfunction: ψ = A r^n sin(n θ); velocity potential: φ = A r^n cos(n θ).
• Setting ψ = 0 at θ = 0 and θ = π/n defines flow in a sector of angle π/n.
• Examples:
  • n = 1: uniform flow (sector angle π, i.e., parallel lines).
  • n = 2: flow into a right-angled corner (sector angle π/2).
  • n = 1/2: wedge flow (sector angle 2π, i.e., flow impinging on a wedge with stagnation at the vertex).

🧮 Rankine half-body (complex form)

• Superimpose uniform flow and a source at the origin: F = U z + (Λ / 2π) ln z.
• Stagnation point: where velocity = 0, solving gives x = −Λ / (2π U), y = 0.
• Streamfunction passing through the stagnation point: ψ_body = 0.
• The ψ = 0 streamline forms a half-body (rounded leading edge, extending to infinity downstream).
• Body width at large r: h = π Λ / U.

🧮 Flow over a cylinder (complex form)

• Uniform flow + doublet: F = U z + μ / z.
• Setting doublet strength μ = U a² (where a is cylinder radius) ensures ψ = 0 at r = a.
• Adding circulation (vortex): F = U z + (U a²) / z − i (Γ / 2π) ln z.
• Stagnation points: sin θ_stag = −Γ / (4π U a).
• Lift force: L = ρ U Γ (Kutta-Joukowsky law).

🔄 Conformal transformations

🔄 What is conformal mapping

Conformal transformation: A mapping function w = f(z) that relates coordinates in one plane (z = x + i y) to another plane (w = ξ + i η), preserving angles and the Laplace equation.

• If F(z) is a complex potential in the z plane, then F(w) is the complex potential in the w plane.
• The velocity in the w plane is related to the velocity in the z plane by: dF/dw = (dF/dz) × (dz/dw).
• Strengths of flow elements (source, vortex, etc.) are unchanged by the transformation.

🔄 Joukowski transformation

Joukowski transformation: w = z + (b² / z), where b is a constant.

• Transforms a circle in the z plane to a different shape in the w plane.
• Example: A circle of radius a = b transforms to a flat plate extending from −2b to +2b along the ξ axis in the w plane.
• Stagnation points in the z plane (on the circle) map to stagnation points in the w plane (on the flat plate).

🔄 Flow over a flat plate at angle of attack

• Uniform flow at angle α to the x axis over a flat plate.
• In the z plane: flow over a circle with circulation adjusted to satisfy the Kutta condition (stagnation point at the trailing edge).
• Required circulation: Γ = 4π U a sin α (where a is circle radius).
• Lift coefficient (nondimensional): C_L = 2π sin α ≈ 2π α (for small α).
• Application: Theoretical lift for a thin (flat) airfoil; more complex airfoil shapes use similar transformations.

🧾 Blasius Theorem

🧾 Force on an arbitrary body

Blasius Theorem: The total force (drag and lift) on a body in inviscid flow is given by the contour integral: D − i L = (ρ / 2) ∮ (dF/dz)² dz, where the integral is around the body surface.

• D is drag (real part, negative sign); L is lift (imaginary part, negative sign).
• For potential flow with no singularities outside the body, the contour can be taken far from the body (in the freestream).
• Far from the body, the complex velocity simplifies (drop terms smaller than 1/z).
• By complex variable theory, the contour integral equals 2π i times the residue (coefficient of the 1/z term in the series expansion of (dF/dz)²).

🧾 Result for rotating cylinder

• Complex potential: F = U z + (U a²) / z − i (Γ / 2π) ln z.
• Complex velocity: dF/dz = U − (U a²) / z² − i (Γ / 2π) / z.
• Squaring and retaining only the 1/z term: residue = −i U Γ / π.
• Force: D − i L = 2π i × (ρ / 2) × (−i U Γ / π) = ρ U Γ.
• Conclusion: Drag D = 0; lift L = ρ U Γ (Kutta-Joukowsky theorem).
• Why it matters: This result holds for any two-dimensional body shape, not just cylinders, as long as the circulation is properly determined.

🧭 Overview

🧠 One-sentence thesis

The panel method is a numerical technique that approximates aerodynamic forces on objects by discretizing the surface into panels, placing flow elements (like vortices) on each panel, and solving for their strengths to satisfy boundary conditions.

📌 Key points (3–5)

• What the method does: approximates inviscid flow solutions by dividing a surface into discrete panels with flow elements that satisfy no-penetration boundary conditions.
• Core mechanism: each panel carries a vortex (or source/doublet) of unknown strength; the sum of induced velocities from all panels must produce zero flow across the surface.
• Key geometric locations: vortex elements are placed at the center of pressure (¼ chord from leading edge), and boundary conditions are checked at collocation points (¾ chord from leading edge).
• Solution process: geometry determines influence coefficients; solving a system of linear equations yields vortex strengths; lift is then calculated from total circulation.
• Common confusion: the method uses potential flow (inviscid), so it captures pressure forces but not viscous drag; it works well for lift prediction before flow separation.

🔧 Method fundamentals

🔧 What the panel method solves

The panel method is an analysis method that can be used to arrive at an approximate solution for the forces acting on an object in a flow, based on inviscid flow analysis, so it is limited to the resultant pressure forces over the surface.

• The governing equation is Laplace's equation for velocity potential.
• Boundary condition: velocity normal to the surface must be zero (impermeable surface).
• Far from the object, flow must equal the freestream velocity.
• The method does not require solving for the entire flow field—only surface velocities are needed to find pressure and force.

🧩 Discretization concept

• The surface is divided into flat panels (straight segments in 2D).
• Each panel carries one flow element (here, a point vortex).
• As panel size decreases, the constructed surface better approximates the real surface.
• Superposition: the total flow at any point is the sum of contributions from all panel elements.

Example: For a wing cross-section, you might use 50 panels along the chord; each panel has its own vortex, and the 50 vortex strengths are unknowns to be solved.

📐 Panel geometry and key locations

📐 Center of pressure and vortex placement

• Center of pressure: the location where the distributed lift force acts with no net moment.
• For a flat panel, thin airfoil theory shows this occurs at ¼ of the panel length from the leading edge.
• Each panel's vortex element is placed at this center of pressure location.
• This choice is convenient and becomes less critical as panels get smaller.

🎯 Collocation points

• Collocation point: the location on each panel where the zero-normal-velocity boundary condition is enforced.
• For a flat panel, this point is at ¾ of the panel length from the leading edge.
• The method only satisfies the boundary condition at this one point per panel (acceptable if panels are reasonably small).

Don't confuse: The vortex location (¼ chord) and collocation point (¾ chord) are different; the vortex induces flow, and the collocation point is where we check that the boundary condition is satisfied.

🧭 Coordinate systems

• Each panel has its own local coordinates: x along the panel, z normal (outward) to the panel.
• The overall system uses a chord-line coordinate system.
• Panel angle α is measured between the panel and the chord line.
• Outward normal vector n points perpendicular to the panel surface.

🔢 Mathematical formulation

🔢 Induced velocity from vortices

The velocity at any point due to a vortex at location (x₀, z₀) with circulation Γ is given by:

• x-component: Γ times (z minus z₀) divided by (2π times distance-squared)
• z-component: negative Γ times (x minus x₀) divided by (2π times distance-squared)

For the panel method:

• Calculate induced velocity at each collocation point from each vortex.
• Use "unit circulation" (Γ = 1) to find influence coefficients.
• Actual velocities are then these coefficients multiplied by the unknown Γ values.

⚖️ Boundary condition equation

At each collocation point j, the sum of induced normal velocities from all vortices plus the freestream normal component must equal zero:

Sum over all panels i of (influence coefficient × Γᵢ) = −U∞ · n̂ⱼ

• Left side: induced flow normal to panel j from all vortices.
• Right side: freestream velocity component normal to panel j (negative because it opposes outward normal).
• This gives N equations for N unknown vortex strengths (N = number of panels).

🧮 Matrix system

The equations can be written in matrix form:

• A · Γ = b
• A is the influence coefficient matrix (depends only on geometry).
• Γ is the vector of unknown vortex strengths.
• b is the freestream contribution vector.

Once geometry is specified, A and b are known, and the system can be solved for Γ.

🎯 Two-panel example

🎯 Simple flat plate setup

Consider a flat plate of chord length c at angle of attack α, divided into two equal panels:

• Panel 1: vortex at (c/8, 0), collocation point at (3c/8, 0).
• Panel 2: vortex at (5c/8, 0), collocation point at (7c/8, 0).
• Both panels have the same angle α (flat plate).
• Both outward normals point in the same direction.

🔧 Influence coefficients

For each collocation point and each vortex, calculate the induced velocity components (u, w) for unit circulation.

• Notation: uᵢⱼ and wᵢⱼ where i = collocation point, j = vortex location.
• Project onto outward normal to get normal component.
• This gives the aᵢⱼ influence coefficients.

📊 Solving for circulation

Write two equations (one per collocation point):

• a₁₁Γ₁ + a₁₂Γ₂ = −U∞ sin(α)
• a₂₁Γ₁ + a₂₂Γ₂ = −U∞ sin(α)

Solve this 2×2 system for Γ₁ and Γ₂.

🚀 Calculating lift

🚀 From circulation to lift

Once vortex strengths are known:

• Lift on each panel: Lᵢ = ρ × U∞ × Γᵢ (Kutta-Joukowski theorem).
• Total lift: sum of all panel lifts.
• Lift is perpendicular to the freestream direction, not necessarily vertical.

📈 Pressure distribution

• Calculate tangential velocity at each panel (induced + freestream tangential component).
• Use Bernoulli equation from freestream to panel surface to find pressure.
• Pressure distribution can be integrated to verify force.

Example: For a wing at 5° angle of attack, solve for all Γᵢ, sum them to get total circulation, multiply by ρU∞ to get lift per unit span.

🌐 Thin airfoil theory overview

🌐 Theoretical foundation

The panel method is based on concepts from thin airfoil theory:

• Assumes foil thickness is much less than chord length.
• Replaces the foil with a vortex sheet along the camber line.
• Continuous distribution of circulation per unit length γ(x).
• Total circulation Γ is the integral of γ(x) along the chord.

🎯 Kutta condition

The Kutta condition states that circulation at the trailing edge must be zero: γ(c) = 0.

• Physically: flow leaves the trailing edge smoothly from top and bottom surfaces.
• No pressure difference at trailing edge → no tendency for flow to swirl.
• This boundary condition allows determination of the circulation distribution.

📐 Lift coefficient from theory

For a thin flat plate at angle of attack α:

• Circulation: Γ = π × c × U∞ × α
• Lift coefficient: Cₗ = 2π × α (for small α in radians)

This simple result shows lift is proportional to angle of attack and provides the theoretical basis for the panel method.

Don't confuse: Thin airfoil theory uses a continuous vortex sheet; the panel method discretizes this into point vortices on panels.

🔍 Extensions and limitations

🔍 Finite span effects

• Real wings have finite span, creating tip vortices.
• Tip vortices induce downwash, effectively reducing angle of attack.
• Aspect ratio AR = span²/area affects lift: Cₗ = (2πα) / (1 + 2/AR)
• Induced drag appears: Cᴅ,induced = Cₗ² / (π × AR)

🏗️ Camber effects

• Camber (curvature of the mean line) adds asymmetry.
• Maximum camber h increases circulation even at zero angle of attack.
• Lift coefficient: Cₗ = 2π(α + 2h/c)
• Flaps effectively increase camber to boost lift.

⚠️ Method limitations

Limitation	Reason	Consequence
Inviscid assumption	No viscous forces included	Cannot predict viscous drag
Before separation	Assumes attached flow	Invalid at high angles of attack
Thin foil	Small perturbation theory	Less accurate for thick airfoils
Pressure forces only	No shear stress	Underpredicts total drag

The panel method works well for lift prediction in the linear range (before stall) but requires corrections or different methods for viscous effects and separated flows.

🧭 Overview

🧠 One-sentence thesis

Viscous flows are governed by the Navier-Stokes equations, which balance acceleration with pressure, body forces, and viscous stresses arising from velocity gradients, and can be solved exactly for many practical flow problems through simplification.

📌 Key points (3–5)

• Viscous forces arise from velocity gradients: friction occurs when adjacent fluid layers move at different speeds, requiring the no-slip boundary condition at surfaces.
• Strain rate vs. rotation rate: the velocity gradient tensor splits into deformation (strain) and rotation (vorticity); viscous stress depends only on strain rate, not rotation.
• Newtonian fluids have linear stress-strain relationships: dynamic viscosity μ is constant with stress (unlike shear-thickening or shear-thinning fluids).
• Common confusion—irrotational vs. inviscid: irrotational flow (zero vorticity) implies inviscid flow, but inviscid flow does not require zero vorticity; viscous terms vanish when curl of vorticity is zero, not just when vorticity is zero.
• Exact solutions exist for simplified cases: fully developed pipe/channel flows and similarity solutions (e.g., suddenly accelerated plate) demonstrate how Navier-Stokes reduces to solvable forms.

🌊 Physical basis of viscous forces

🧱 No-slip boundary condition and velocity profiles

• No-slip condition: fluid in contact with a surface takes on the surface velocity—no "slipping" allowed.
• This creates velocity gradients normal to the surface.
• Internal flows (pipes): velocity is zero at the wall, maximum at the centerline; symmetric profile for circular pipes.
• External flows (flat plates): velocity transitions from zero at the surface to freestream velocity far away; the friction layer grows along the flow direction.
• Example: in a stationary circular pipe with flow, the center moves fastest because it is farthest from wall friction; fluid-fluid friction occurs throughout due to different "layers" moving at different speeds.

🔧 Viscous stress tensor τ

Viscous stress tensor: a nine-element tensor (three diagonal, six off-diagonal) representing stresses on a fluid element caused by velocity gradients.

• Diagonal terms (i = j): normal stresses causing linear deformation (stretching/compression).
• Off-diagonal terms (i ≠ j): shear stresses causing angular deformation.
• The tensor is symmetric: τ_ij = τ_ji.
• Viscous forces act at surfaces and depend on velocity gradients; where gradients vanish, viscous forces vanish.

📐 Strain rate and deformation

📏 Linear strain rate (dilation)

• Linear deformation: when velocity u increases in the x direction, the right face of a fluid element moves faster than the left, stretching the element.
• Rate of linear strain in x direction: ∂u/∂x (similarly ∂v/∂y and ∂w/∂z for y and z).
• Dilation rate: sum of all three linear strain rates, ∂u_i/∂x_i (summation implied), measures volume change per time per volume.
• Units: one over time.

🔄 Angular strain rate (shear)

• Shear deformation: velocity gradients cause rotation of line segments within the element.
• Example: if u increases in the y direction, points A-B rotate about the z axis at rate ∂u/∂y.
• Similarly, ∂v/∂x also causes rotation about z.
• The strain rate tensor e_ij combines both: e_ij = (1/2)(∂u_i/∂x_j + ∂u_j/∂x_i).
• The factor 1/2 provides an average of the two contributions.
• The tensor is symmetric (e_ij = e_ji) and has six independent components.

🌀 Distinguishing strain rate from vorticity

• Vorticity measures net rotation rate: ω_z = ∂v/∂x − ∂u/∂y (note the sign difference from strain rate).
• Vorticity sums contributions of the same rotation direction; it is a vector (three components).
• Key distinction: zero vorticity means no net rotation, but angular deformation can still be nonzero.
• Example: if ∂v/∂x = ∂u/∂y, vorticity is zero but angular strain rate is 2(∂u/∂y), causing element distortion without rotation.
• The velocity gradient tensor splits: ∂u_i/∂x_j = e_ij + (1/2)ω_ij, where e_ij is strain rate and ω_ij is rotation rate.

🧪 Constitutive model and fluid types

🔗 Stress-strain rate relationship

Newtonian fluid model: viscous stress is proportional to strain rate, τ_ij = 2μ e_ij, where μ is dynamic viscosity.

• Viscous stress depends on deformation rate, not rotation rate.
• The proportionality factor 2μ makes the math convenient.
• Normal stresses (i = j): linked to linear deformation (volume change).
• Shear stresses (i ≠ j): linked to angular deformation (shape distortion).
• Dynamic viscosity μ is a fluid property (SI units: Pa·s or kg/(m·s)).

🧴 Newtonian vs. non-Newtonian fluids

Fluid type	Viscosity behavior	Stress-strain relationship	Examples
Newtonian	Constant with stress (may vary with temperature/pressure)	Linear	Water, air, oil
Dilatant (shear-thickening)	Increases with stress	Nonlinear, slope increases	Cornstarch-water mixture
Pseudoplastic (shear-thinning)	Decreases with stress	Nonlinear, slope decreases	Ketchup, paint, ink

• Don't confuse: a Newtonian fluid can have temperature-dependent viscosity (like oil) but still has constant viscosity at any given stress level.
• This text focuses only on Newtonian fluids.

📜 Navier-Stokes equation derivation

⚖️ Force balance and stress components

• Euler equation (inviscid): ρ(∂u_i/∂t + u_j ∂u_i/∂x_j) = −∂P/∂x_i + ρg_i.
• Total stress: σ_ij = −P δ_ij + τ_ij, where δ_ij is the Kronecker delta (1 if i = j, else 0); pressure is a compressive normal stress (negative sign).
• Bulk viscosity term: accounts for forces resisting volume change, proportional to dilation rate; for incompressible flow this term vanishes (dilation rate is zero).
• Stokes hypothesis: for compressible flow, λ = −(2/3)μ, where λ is bulk viscosity coefficient; valid for most conditions but not extreme pressures.

🧮 Final incompressible form

Navier-Stokes equation (incompressible, constant viscosity): ρ(∂u_i/∂t + u_j ∂u_i/∂x_j) = −∂P/∂x_i + ρg_i + μ ∂²u_i/∂x_j∂x_j.

• Left side: local acceleration + convective acceleration (mass times acceleration per volume).
• Right side: pressure gradient force + gravity body force + viscous force (normal and shear).
• Coupled with continuity: ∂u_i/∂x_i = 0 (incompressible).
• The viscous term simplifies to μ times the Laplacian of velocity (∇²u) for constant μ and incompressible flow.
• Conservative form: convective term can be written as ∂(u_i u_j)/∂x_j using continuity.

🔁 Vorticity form and irrotational flow

• Using vector identities, convective acceleration: u_j ∂u_i/∂x_j = ∂(u·u/2)/∂x_i − u_j ω_ij.
• Viscous term: μ ∂²u_i/∂x_j∂x_j = μ ∂ω_j/∂x_i (curl of vorticity).
• Irrotational flow (ω = 0): convective term simplifies and viscous term vanishes → recovers Bernoulli equation when integrated.
• Inviscid flow (μ = 0): viscous term is zero, but vorticity may still be nonzero (enters via convective term).
• Key insight: irrotational implies inviscid, but inviscid does not imply irrotational; viscous term also vanishes if curl of vorticity is zero (not just if vorticity is zero).

🔬 Exact solutions to Navier-Stokes

🚰 Fully developed flow between parallel plates

• Setup: steady flow in x direction between plates separated by distance 2h; y = 0 at centerline.
• Assumptions: fully developed (∂u/∂x = 0), steady (∂u/∂t = 0), no acceleration.
• Governing equation reduces to: 0 = −∂P/∂x + μ ∂²u/∂y² (horizontal) or 0 = ρg + μ ∂²u/∂y² (vertical gravity-driven).
• Pressure gradient ∂P/∂x is constant (not a function of y or z).
• Solution (parabolic profile): u(y) = −(1/(2μ)) (dP/dx or ρg) (h² − y²), where dP/dx is constant pressure drop per length.
• Boundary condition: u = 0 at y = ±h (no-slip).
• Maximum velocity at centerline (y = 0).
• Vorticity is piecewise linear, zero at center, maximum at walls.

🔵 Hagen-Poiseuille flow (circular pipe)

• Setup: steady, horizontal, constant-area pipe of radius R; flow in axial (z) direction.
• Governing equation: same force balance in cylindrical coordinates, 0 = −∂P/∂z + μ (1/r) ∂/∂r(r ∂u_z/∂r).
• Pressure gradient is constant along z.
• Solution (parabolic): u_z(r) = −(1/(4μ)) (dP/dz) (R² − r²).
• Boundary conditions: u_z = 0 at r = R; ∂u_z/∂r = 0 at r = 0 (maximum at centerline).
• Volumetric flow rate: Q = −(π R⁴)/(8μ) (ΔP/L), where ΔP is upstream minus downstream pressure over length L.
• Similar result for parallel plates: Q = −(2h³ W)/(3μ) (ΔP/L), where W is plate width.

🏃 Suddenly accelerated plate (Stokes first problem)

• Setup: infinite flat plate in infinite fluid, plate accelerates instantly to velocity U₀ at t = 0; flow only in x direction, varies with y and t.
• Governing equation: ∂u/∂t = ν ∂²u/∂y², where ν = μ/ρ is kinematic viscosity.
• This is a diffusion problem: momentum diffuses away from the wall.
• Boundary/initial conditions: u = U₀ at y = 0 (no-slip); u → 0 as y → ∞; u = 0 at t = 0.

🔢 Similarity solution method

• Dimensional analysis: u depends on U₀, y, t, ν → nondimensional groups u/U₀ and y/√(νt).
• Similarity variable: η = y/(2√(νt)) collapses two independent variables (y, t) into one.
• Length scale √(νt) grows with time (diffusion distance increases).
• Transforms PDE in (y, t) into ODE in η: d²f/dη² + 2η df/dη = 0, where f = u/U₀.
• Solution: f(η) = 1 − erf(η), where erf is the Gaussian error function (ranges from −1 to +1, zero at η = 0).
• Boundary conditions: f(0) = 1, f(∞) = 0.

📊 Physical interpretation

• Velocity profile: u/U₀ = 1 − erf(y/(2√(νt))); decays from plate velocity to zero far away.
• Wall shear stress: τ_wall = μ ∂u/∂y at y = 0 = μ U₀/(√(π ν t)); decreases with time as momentum diffuses.
• Force on plate: F = τ_wall × (area) = μ U₀ L S/(√(π ν t)), where L is length, S is width.
• Diffusion distance: δ ≈ 3.7√(νt) (distance where u drops to 1% of U₀); grows with √t and √ν.
• Greater viscosity → faster diffusion (counterintuitive: higher μ spreads momentum farther).
• Vorticity: ω_z = −∂u/∂y, concentrated near surface, spreads over time.
• Circulation: Γ = U₀ (constant over time); total vorticity is conserved, just spread over increasing area.
• Example: this models start-up of a lubricated plate, valid until flow reaches surrounding boundaries.

🧭 Overview

🧠 One-sentence thesis

Boundary layer theory explains how viscous friction near a surface creates a thin region where velocity transitions from zero at the wall to the freestream value, with thickness growing downstream and shear stress decreasing along the flow direction.

📌 Key points (3–5)

• What a boundary layer is: a thin region next to a surface where wall friction significantly slows the flow, while fluid outside this layer remains unaffected by friction.
• How it grows: boundary layer thickness increases downstream as momentum diffusion spreads friction effects further from the surface; thickness scales as the square root of distance divided by the square root of Reynolds number.
• Why wall-normal velocity exists: conservation of mass requires an upward velocity component because the slowed boundary layer flow reduces downstream mass flux.
• Common confusion—pressure gradient: for a flat surface the pressure gradient along the flow is essentially zero inside the boundary layer (same as outside), but curved surfaces or adverse pressure gradients can cause flow separation.
• Laminar solution (Blasius): the flat-plate boundary layer can be solved as a similarity equation, yielding velocity profiles and surface drag that depend on Reynolds number.

🌊 Physical nature of boundary layers

🌊 Definition and region of influence

A boundary layer flow is the region of a larger flow field next to the surface that has significant effects of wall frictional forces.

• The no-slip boundary condition forces velocity to zero at the surface.
• Friction diffuses momentum loss in the direction normal to the surface, reducing local fluid velocity.
• Outside this thin layer, the fluid is directly unaffected by wall friction; the velocity there is called the "freestream" value, U.
• The boundary layer thickness δ increases along the flow direction as more fluid is slowed by friction.

📏 Growth and diffusion

• The rate of diffusion depends on fluid viscosity.
• As the flow moves downstream, slower-moving fluid near the surface exerts frictional forces on fluid further away.
• The region slowed by friction grows, so δ increases with downstream distance x.
• Example: at the leading edge (x = 0), the boundary layer just begins to form; further downstream, δ is larger because friction has had more time to diffuse momentum loss outward.

🔼 Wall-normal velocity component

• Although flow is principally along the surface (x direction), there must also be a nonzero velocity component v in the y direction (normal to the wall).
• Why: applying conservation of mass to a control volume from the leading edge to distance x shows that the slowed velocity profile at x carries less mass flux than the uniform inflow at the leading edge.
• Consequently, mass must exit through the top of the control volume, requiring v > 0.
• The no-slip condition requires v = 0 at the surface, so v starts at zero and increases with increasing y.
• Streamlines deflect upward along the flow direction to account for this nonzero v component.

📉 Surface shear stress distribution

• As δ grows downstream, the velocity at any given height y must decrease in the x direction (∂u/∂x < 0 within the boundary layer).
• The velocity gradient ∂u/∂y at the surface therefore decreases along x.
• Surface shear stress τ_w = μ (∂u/∂y) at y = 0, so τ_w decreases in the x direction.
• Don't confuse with pipe flow: in boundary layer flow, shear stress never reaches a constant value because the flow never becomes fully developed (δ keeps growing).

🧮 Governing equations and scaling

🧮 Simplifications from physical attributes

The boundary layer equations are a simplified form of the Navier-Stokes equations based on four key attributes:

Attribute	Implication
Thin boundary layer	δ << x (or δ << L)
Small wall-normal velocity	v << u
Flat surface	Pressure gradient ∂p/∂x ≈ 0
Outer edge condition	u → U (freestream) at edge of boundary layer

• For a flat surface, the pressure gradient along the flow is essentially zero inside the boundary layer because it is zero just outside the boundary layer.
• Even if hydrostatic pressure varies across δ, the change in pressure along x is the same at the edge and near the surface, so ∂p/∂x ≈ 0 everywhere in the boundary layer.

📐 Nondimensionalization and scaling

The x-momentum Navier-Stokes equation for steady, two-dimensional, incompressible flow over a flat plate (neglecting gravity and pressure gradient):

u (∂u/∂x) + v (∂u/∂y) = ν (∂²u/∂x² + ∂²u/∂y²)

Combined with conservation of mass:

∂u/∂x + ∂v/∂y = 0

Characteristic scales:

• x ~ L (surface length)
• y ~ δ (boundary layer thickness at x = L)
• u ~ U (freestream velocity)
• v ~ ? (to be determined)

Finding the v scale: inserting nondimensional variables into conservation of mass and requiring both terms to be order one yields:

v ~ U (δ / L)

This shows v is smaller than u by the ratio δ/L, consistent with the thin boundary layer assumption.

🔬 Order of magnitude analysis

Nondimensionalizing the momentum equation and examining the viscous terms:

• The term containing ∂²u/∂y² has coefficient 1 / Re_L, where Re_L = U L / ν.
• The term containing ∂²u/∂x² has coefficient (δ/L)² / Re_L.
• Since δ << L, the second term is much smaller and can be neglected.
• Conclusion: frictional forces in the boundary layer are dominated by shearing stress (∂²u/∂y²), not normal viscous stress (∂²u/∂x²).

📊 Reynolds number and boundary layer thickness

For the equation to balance (all terms order one), the coefficient 1 / Re_L must also be order one, implying:

δ / L ~ 1 / √(Re_L)

or

δ ~ L / √(Re_L)

Key implications:

• For large Reynolds numbers, a thin boundary layer exists.
• For very small Re_L, boundary layer conditions do not hold (δ is not small).
• Replacing L with any position x along the surface: δ(x) ~ x / √(Re_x), where Re_x = U x / ν.
• Boundary layer grows proportional to √x (since δ ~ √(x / (U/ν)) = √(ν x / U)).

📝 Final boundary layer equation (nondimensional, flat plate)

After scaling and neglecting small terms:

u* (∂u*/∂x*) + v* (∂u*/∂y*) = ∂²u*/∂y*²

with boundary conditions:

• u* = 0 at y* = 0 (no-slip)
• u* → 1 as y* → ∞ (freestream)
• u* = 1 at x* = 0 (leading edge, outside boundary layer)

This is a parabolic partial differential equation requiring only one boundary condition in x (at the leading edge), so the solution is independent of downstream conditions and can be solved at any arbitrary position x.

🎯 Blasius solution for laminar flow

🎯 Streamfunction approach

To solve the boundary layer equations, introduce the streamfunction ψ:

u = ∂ψ/∂y
v = -∂ψ/∂x

This automatically satisfies conservation of mass.

Substituting into the dimensional momentum equation:

(∂ψ/∂y)(∂²ψ/∂x∂y) - (∂ψ/∂x)(∂²ψ/∂y²) = ν (∂³ψ/∂y³)

🔄 Similarity transformation

Define a nondimensional streamfunction and similarity variable:

ψ = √(ν U x) f(η)
η = y √(U / (ν x))

where f is a function of η only.

After substituting and applying the chain rule, the result is a single-variable ordinary differential equation:

f''' + (1/2) f f'' = 0

where primes denote derivatives with respect to η.

Boundary conditions:

• f(0) = 0 and f'(0) = 0 (no-slip: u = v = 0 at y = 0)
• f'(η) → 1 as η → ∞ (u → U at edge of boundary layer)

This is a similarity equation, meaning the velocity profile shape is the same at all x locations when plotted against η.

📈 Solution results

The numerical solution (originally by Paul Blasius around 1913) gives f'(η) as a function of η.

Boundary layer thickness: choosing f' = 0.99 (99% of freestream velocity) occurs at η ≈ 5.0, so:

δ = 5.0 √(ν x / U) = 5.0 x / √(Re_x)

This confirms the scaling δ ~ x / √(Re_x) with the constant C = 5.0.

Velocity components: once f(η) is known, u and v can be evaluated at any (x, y) location in the boundary layer using the definitions of ψ and η.

🧱 Surface shear stress and drag

The surface shear stress:

τ_w = μ (∂u/∂y) at y = 0

Evaluating the derivative using the similarity solution gives:

τ_w = 0.332 μ U √(U / (ν x))

The local skin friction coefficient:

C_f = τ_w / (½ ρ U²) = 0.664 / √(Re_x)

Total drag force: integrating τ_w over the entire surface (length L, span S):

Drag coefficient: C_D = 1.328 / √(Re_L)

where Re_L = U L / ν.

Key result: surface shear stress decreases as 1/√x, and total drag scales as √L (or √(Re_L)).

🌀 Pressure gradient effects and flow separation

🌀 Pressure and friction coupling

For viscous flows, pressure and surface frictional forces cannot be treated independently:

• The pressure distribution influences the velocity distribution (and vice versa) through the Navier-Stokes equation.
• An altered velocity profile changes the velocity gradient at the surface, changing the surface stress.

⚖️ Navier-Stokes at the surface

Evaluating the momentum equation at the surface (where u = v = 0, so acceleration terms vanish):

∂p/∂x = μ (∂²u/∂y²) at y = 0

(Here p includes the hydrostatic contribution: p + ρ g h.)

This shows the link between pressure gradient and the curvature of the velocity profile at the surface.

➕ Favorable pressure gradient

Favorable pressure gradient: ∂p/∂x < 0 (pressure decreases in the flow direction).

• From the surface equation, ∂²u/∂y² < 0, meaning stress decreases moving away from the surface (as expected, since stress → 0 at the edge of the boundary layer).
• For very large favorable gradients, the boundary layer becomes thinner and wall shear stress becomes larger.
• Example: upstream side of a cylinder, where flow accelerates and pressure drops.

➖ Adverse pressure gradient

Adverse pressure gradient: ∂p/∂x > 0 (pressure increases in the flow direction).

• From the surface equation, ∂²u/∂y² > 0, meaning stress increases with increasing y away from the surface.
• Stress reaches a maximum somewhere in the boundary layer, then decays to zero at the edge.
• As the adverse gradient increases, the stress gradient near the surface becomes larger, forcing the stress to approach zero or even go negative near the wall.
• Negative stress implies flow reversal: fluid near the wall moves upstream while fluid further away moves downstream.

🌪️ Flow separation

Flow separation: when the adverse pressure gradient is large enough, the flow near the surface reverses direction (negative stress), and the flow no longer adheres to the surface.

• Separated flow results in swirling, turbulent flow near the surface (a "wake").
• The flow pattern becomes asymmetric.
• Common cause: large surface curvature, such as the downstream side of a cylinder.

Example—flow over a cylinder:

• On the upstream side, pressure gradient is negative (favorable); flow accelerates, boundary layer is thin.
• At the top/bottom of the cylinder, velocity is maximum, pressure is minimum.
• On the downstream side, pressure gradient is positive (adverse); flow decelerates, boundary layer thickens.
• For sufficiently high freestream velocity (high Re), the adverse gradient causes separation, forming a wake across the back of the cylinder.
• Inviscid case: pressure distribution is symmetric, net pressure force is zero.
• Viscous case: wake pressure is lower than inviscid prediction, pressure recovery on downstream side is incomplete, resulting in a net downstream drag force.
• Turbulent flows tend to have greater pressure recovery and lower drag coefficient than laminar flows.

📊 Pressure coefficient

For inviscid flow over a cylinder, the pressure coefficient:

C_p = (p - p_∞) / (½ ρ U²)

where p_∞ is the upstream pressure and the surface velocity varies around the cylinder (zero at stagnation points, maximum at top/bottom).

With viscous effects and separation, the pressure distribution is no longer symmetric, and the drag coefficient C_D (based on frontal area) is nonzero.

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Note: The excerpt mentions that the integral boundary layer method (Chapter IX) takes the opposite approach—integrating the governing equations across the flow and solving for wall shear stress as an unknown, with pressure gradient as a known parameter—but does not provide details of that method.

🧭 Overview

🧠 One-sentence thesis

The integral boundary layer method provides a powerful tool to evaluate surface viscous forces by integrating the governing equations across the flow with wall shear stress as an unknown to be solved for, rather than solving directly for the velocity profile as in the Blasius approach.

📌 Key points (3–5)

• Core approach difference: The integral method integrates equations across the flow to solve for wall shear stress, opposite to the Blasius method which first finds the velocity profile then calculates surface stress from it.
• Pressure gradient handling: The local pressure gradient is treated as a known parameter at each position, assuming the freestream pressure gradient equals the boundary layer pressure gradient (valid for both laminar and turbulent flows).
• Three key scaling parameters: Boundary layer thickness (δ), displacement thickness (δ*), and momentum thickness (θ) characterize the flow and help predict surface force distributions.
• Common confusion: Displacement and momentum thicknesses are not the actual boundary layer thickness—they measure mass flow deficit and momentum deficit respectively, and are typically less than δ.
• Solution method: Assumes a nondimensional velocity profile satisfying boundary conditions, then evaluates integrals to find surface stress without needing the precise surface velocity derivative.

📐 Scaling Parameters

📏 Boundary layer thickness (δ)

The boundary layer thickness δ represents the thickness measured from the surface to the location where the velocity reaches the freestream value U.

• Since velocity asymptotically approaches the freestream value, δ is often defined as where velocity reaches 99% of U (1% criterion).
• How pressure gradients affect δ:
  • Favorable pressure gradient (pressure decreases along flow): thinner boundary layer, higher surface shear stress
  • Adverse pressure gradient (pressure increases along flow): thicker boundary layer, lower surface shear stress
• δ is a function of downstream coordinate x and increases in the x direction.
• Can be expressed through Reynolds number since Re depends on x: δ increases as Re increases.

📉 Displacement thickness (δ*)

Displacement thickness δ* represents the distance the surface would need to be moved upward (in the y direction) to capture all of the lost mass flow rate if the velocity were uniformly at U (frictionless).

Definition:

• δ* = integral from 0 to δ of (1 - u/U) dy
• Based on mass flow rate deficit caused by surface friction

Physical interpretation:

• Surface friction causes velocity to decrease, creating a mass flow deficit within the boundary layer
• The deficit equals the product of density ρ, freestream velocity U, span S, and the "thickness" δ*
• If total mass flow rate lost due to friction is divided by ρU·S, the result is δ*

Behavior along the flow:

• δ* increases in the downstream direction as more fluid is slowed by friction
• Greater local surface shear stress → larger relative value of δ*
• Favorable pressure gradient increases local shear stress → larger relative δ* compared to zero pressure gradient

Don't confuse: δ* is not the actual boundary layer thickness; it is typically less than δ and measures mass flow deficit, not the physical extent of the boundary layer.

🎯 Momentum thickness (θ)

Momentum thickness θ represents the distance the surface would need to be moved into the flow (in the y direction) if velocity is uniformly at U in order to reduce the momentum rate equivalent to the actual rate of momentum loss.

Definition:

• θ = integral from 0 to δ of (u/U)(1 - u/U) dy
• Based on momentum rate deficit caused by friction

Physical interpretation:

• Momentum rate lost crossing a vertical plane = integral of (mass flow rate) × (momentum lost per mass flow rate)
• The expression (U - u) represents momentum lost per mass flow rate
• Setting momentum rate lost equal to ρU²·S·θ and solving gives the definition

Relationship to surface stress:

• For a flat surface with zero pressure gradient: τ_w = ρU² (dθ/dx)
• This shows that the rate of change of momentum thickness decreases along the flow direction since surface stress decreases downstream
• If θ can be determined, it is possible to find the wall surface stress τ_w

Don't confuse: θ measures momentum deficit, not mass deficit (that's δ*); both are less than the actual boundary layer thickness δ.

🧮 General Integral Boundary Layer Equations

🔧 Derivation starting point

The derivation begins with the Navier-Stokes equation simplified for boundary layer flow:

• ∂u/∂t + u(∂u/∂x) + v(∂u/∂y) = -(1/ρ)(∂P/∂x) + ν(∂²u/∂y²)
• Where ν is kinematic viscosity

Key assumption: The pressure gradient term is determined from the freestream flow, assumed inviscid, using Euler's equation:

• (1/ρ)(∂P/∂x) = -U(∂U/∂x)
• This connects boundary layer pressure gradient to freestream velocity changes

🔄 Integration process

The equation is integrated across the boundary layer (in the y direction) for a given value of x, with boundary conditions:

• No-slip at y = 0
• u = U at y = δ

Continuity equation role:

• Two-dimensional continuity: ∂u/∂x + ∂v/∂y = 0
• Integrated from 0 to δ: v(δ) = -integral from 0 to δ of (∂u/∂x) dy
• This expression is used in the second term of the momentum equation

Integration by parts technique:

• Applied to the convective term v(∂u/∂y)
• Uses boundary conditions to simplify the result

📝 Final integral form

After combining all terms and applying definitions of δ* and θ, the general integral boundary layer equation is:

With pressure gradient:

• d/dx(U²θ) + δ*U(dU/dx) = τ_w/ρ

For flat plate (U = constant):

• dθ/dx = τ_w/(ρU²) = C_f/2
• Where C_f is the skin friction coefficient

What this achieves:

• Reduces the problem to a differential equation in terms of θ and δ* (both functions of x)
• U(x) is determined by the pressure gradient through Euler's equation
• The surface stress distribution can be found by solving this equation

🛠️ Solution Method

🎨 Assumed velocity profile approach

The method assumes a nondimensional velocity profile rather than solving for it directly.

Nondimensional variable:

• η = y/δ (where δ is a function of x)
• This scaling combines cross-stream and stream-wise dependency

Assumed form:

• u/U = f(η)
• Example: third-order polynomial u/U = a₀ + a₁η + a₂η² + a₃η³

Key insight: Since surface stress depends on the integral of the velocity distribution (through θ and δ*), good values for the integrals are needed rather than the precise surface derivative of velocity.

🎯 Boundary conditions

For a polynomial velocity distribution, coefficients are determined by boundary conditions:

1. At the surface (η = 0): u/U = 0 (no-slip)
2. At boundary layer edge (η = 1): u/U = 1 (velocity reaches freestream)
3. At boundary layer edge (η = 1): ∂(u/U)/∂η = 0 (no shear)
4. At boundary layer edge (η = 1): ∂²(u/U)/∂η² = 0 (no net viscous force)

For third-order polynomial, these four conditions yield:

• u/U = (3/2)η - (1/2)η³

Don't confuse: Higher-order polynomials require more boundary conditions; the fourth condition comes from the Navier-Stokes equation at the boundary layer edge.

🔢 Calculation procedure

Step-by-step process:

1. Assume a functional form for u/U = f(η)
2. Match required boundary conditions to find coefficients
3. Calculate the constant δ*/δ from the integral definition
4. Calculate the constant θ/δ from the integral definition
5. Calculate δ(x), which determines the local skin friction coefficient C_f

Transforming to nondimensional variables:

• δ* = δ × integral from 0 to 1 of [1 - f(η)] dη
• θ = δ × integral from 0 to 1 of f(η)[1 - f(η)] dη
• Both δ*/δ and θ/δ become constants once f(η) is specified

Differential equation for δ:

• For flat plate: dθ/dx = (θ/δ)(dδ/dx) = τ_w/(ρU²)
• Using τ_w = μU(∂u/∂y) at y=0 and transforming to η variables
• Results in: δ dδ = (constant dependent on polynomial order) × (ν/U) dx

Integration result:

• δ² = 2C₁(νx/U)
• Where C₁ is a constant dependent on the polynomial order

📊 Results for third-order polynomial

For the velocity profile u/U = (3/2)η - (1/2)η³:

• δ*/δ = 3/8
• θ/δ = 39/280
• C₁ = 140/13
• δ = 4.64x/√(Re_x)
• C_f = 0.646/√(Re_x)

Total friction drag coefficient:

• C_D = (Drag force)/(½ρU²·Area) = 2 × C_f evaluated at x = L
• C_D = 1.292/√(Re_L)

Accuracy: Results compare very closely with experimental data for appropriate Reynolds number conditions, making this a fairly easy yet surprisingly accurate method to predict friction drag forces on flat surfaces.

🌀 Pressure Gradient Complications

⚠️ Challenge with curved surfaces

When pressure gradients are present (surfaces with curvature), the solution becomes more challenging:

• At the surface, net viscous forces are not zero but are balanced by the pressure gradient
• Velocity profiles need to reflect this condition in the boundary conditions
• The fourth boundary condition (∂²u/∂y² = 0 at the edge) no longer applies at the surface

Consequence: More sophisticated procedures are needed to obtain a good velocity profile that satisfies the modified boundary conditions.

Don't confuse: The flat plate case (zero pressure gradient) is a special simplification; most real flows involve pressure gradients that significantly affect the boundary layer development and require different boundary conditions for the assumed velocity profile.

🧭 Overview

🧠 One-sentence thesis

Turbulent flows exhibit complex spatio-temporal fluctuations across a broad spectrum of scales that significantly alter mean momentum distribution and increase wall shear stress compared to laminar flows, requiring empirical models and Reynolds averaging to incorporate turbulence effects into engineering analyses.

📌 Key points (3–5)

• What turbulence is: time-varying fluctuations of all dependent variables (velocity components and pressure) occurring across multiple spatial and temporal scales simultaneously.
• How turbulence affects momentum: Reynolds averaging reveals that turbulent fluctuations create additional "Reynolds stress" terms that flatten velocity profiles and increase wall shear stress.
• Spectrum of scales: turbulent flows contain large-amplitude, low-frequency fluctuations superimposed with small-amplitude, high-frequency fluctuations, spanning several orders of magnitude in energy.
• Common confusion: turbulent stress vs. viscous stress—in many situations turbulent stress dominates except very near surfaces where fluctuations are dampened.
• Practical consequence: turbulent boundary layers have flatter velocity profiles but higher velocity gradients near walls, resulting in larger surface friction forces than laminar flows.

🌊 Physical characteristics of turbulence

🌊 Spatio-temporal fluctuations

Turbulence is typically described as a time varying phenomena of all dependent variables (except density for incompressible flows). In addition, spatial variations of the time dependency occurs as well.

• All velocity components and pressure vary in both time and space.
• The Navier-Stokes equations are nonlinear through convective acceleration terms, so statistical descriptions rely on correlation functions among variables.
• Flow geometry impacts larger scales (which interact with boundaries), but smaller scales tend to be independent of geometry, suggesting underlying theoretical dynamics.

📊 Energy spectrum and scales

• A typical velocity signal shows fluctuations at many different time scales: small-scale (short) events with small amplitudes coupled with longer, higher-amplitude fluctuations.
• Energy spectrum: Fourier transformation decomposes mean-square fluctuations by frequency.
  • Larger amplitude fluctuations occur at longer time scales (lower frequency).
  • Smaller amplitude fluctuations occur at shorter time scales (higher frequency).
  • Energy difference spans several orders of magnitude between largest and smallest scales.
• Don't confuse with: a single oscillating frequency would appear as a spike in the energy spectrum and would not be turbulent.

Example: The short time events appear superimposed over longer events in the velocity signal, reflecting this broad spectrum.

🔥 Turbulent kinetic energy (TKE)

The integration of the energy spectrum over all frequencies yields total fluctuation energy.

Turbulent intensity (for one velocity component):

• Turbulent intensity = square root of (time average of velocity fluctuation squared) divided by mean velocity.
• The fluctuating velocity is the instantaneous value minus the time average value.

Turbulent kinetic energy (TKE):

• Sum of mean-square values for all three velocity components.
• Greater TKE typically means more intense turbulence.
• Can be measured at each location to determine spatial distribution of turbulence in a time-averaged sense.

🧮 Reynolds averaging and Reynolds stress

🧮 Reynolds decomposition

The instantaneous velocity can be expressed as the sum of the fluctuating part and the mean value at any point in the flow.

• Instantaneous velocity = mean value + fluctuating part (primed quantity, time-dependent).
• The time average of the fluctuating part is zero by definition.
• Each time-dependent term in continuity and Navier-Stokes equations is replaced with mean plus fluctuating components, then time-averaged.

⚙️ Convective acceleration and correlation

For the convective acceleration term after Reynolds decomposition and time averaging:

• Terms containing products of mean and fluctuating parts average to zero.
• The convective acceleration using time-average velocities remains.
• Key new term: time average of convective acceleration of fluctuating velocities does not equal zero.
  • The average of products of time-varying signals ≠ product of averages of those signals.

For incompressible flow, this term becomes the spatial derivative of the correlation function (time-averaged product of velocity components):

• This is a second-order symmetric tensor with six independent components.
• Represents the contribution of turbulence to mean momentum of the flow.

🔧 Reynolds stress definition

"Reynolds stress": the turbulent stress term that modifies the momentum distribution in the flow.

The turbulent stress is written as:

• Turbulent stress = negative of (time-averaged product of fluctuating velocity components).
• Named in honor of Reynolds, who developed the decomposition.

Total stress = viscous stress + turbulent stress.

• In many situations, viscous stress is much smaller than turbulent stress and can be ignored except near surfaces where fluctuations are dampened.
• Don't confuse: the Reynolds stress is not a physical viscous force but a statistical effect of turbulent momentum transport.

🏔️ Turbulent boundary layers

🏔️ Governing equations and assumptions

The discussion assumes turbulent flow over the entire surface (conditions discussed later).

Assumptions (same as laminar boundary layer):

• Thin boundary layer: boundary layer thickness much less than streamwise length.
• Streamwise velocity greater than cross-stream velocity.
• Cross-stream derivatives greater than streamwise derivatives.
• Large Reynolds number.

The time-averaged momentum equation for constant property flows (ignoring body force):

• Includes pressure gradient term (only mean pressure remains after averaging).
• Key observation: in the Reynolds stress term, only the cross-correlation (product of streamwise and cross-stream fluctuating velocities) contributes, even though both fluctuating components may be similar magnitude.

🔄 Physical mechanism of Reynolds stress

The cross-correlation in Reynolds stress impacts mean momentum distribution and wall shear stress.

How it works (referring to Fig. 10.3):

• Positive upward fluctuation moves low-velocity fluid into higher-velocity region → instantaneous streamwise fluctuation becomes negative.
• Negative downward fluctuation moves high-velocity fluid into slower region → again, streamwise fluctuation becomes negative.
• On average, the cross-correlation is negative, so the added turbulent stress is positive.

Consequence:

• Added stress slows down higher-speed fluid and speeds up lower-speed fluid.
• Result: much flatter velocity/momentum distribution across the flow.
• No-slip condition still holds, so flatter profile implies higher velocity gradients near surface.
• Net result: turbulent boundary layer has larger surface viscous stresses than laminar.

📏 Scaling and velocity profiles

Friction velocity (new velocity scale proportional to wall stress):

• Friction velocity = square root of (wall shear stress divided by density).
• Not a specific velocity in the flow, but has velocity units and is proportional to velocity gradient at surface.
• Reasonable to note friction velocity is proportional to freestream velocity.

Power law velocity distribution:

• Nondimensional velocity = (cross-stream coordinate / boundary layer thickness) to the power of (1/n).
• n is a positive integer; for moderately large Reynolds numbers, n ≈ 7 matches data well.
• As Reynolds number increases, velocity profile becomes flatter and n increases.

Displacement and momentum thickness: defined identically as laminar flow but using time-average velocities.

🧱 Wall shear stress and drag calculations

🧱 Skin friction relationships

An alternative scaling with power law assumption relates friction velocity to boundary layer thickness.

For a flat plate, the momentum integral equation (valid for both laminar and turbulent) yields:

• Skin friction coefficient depends on Reynolds number based on boundary layer thickness.
• Based on experimental data analyzed by Blasius for turbulent flow, constants are determined.

For n = 7:

• Skin friction coefficient = 0.027 / (Reynolds number based on streamwise distance) to the power of (1/7).

Total drag force: obtained by integrating surface shear stress over area (length L, width S).

🚀 Initial laminar starting length

Turbulent flow over a surface generally does not exist from the leading edge (x = 0).

Transition criterion:

• Experiments show turbulent flow on smooth surface occurs when Reynolds number (based on streamwise distance) > 5×10⁵.
• Reynolds number is zero at leading edge, so laminar region exists near leading edge, then flow becomes turbulent further along if Reynolds number is large enough.

Tripped flow: turbulence can occur from or very close to leading edge if slight roughness exists and freestream velocity is sufficiently large.

Solution approach:

• Divide integration of surface shear stress into laminar and turbulent regions.
• Transition location found using critical Reynolds number.
• Total drag coefficient accounts for both regions by adding/subtracting contributions.

Result (assuming transition at critical Reynolds number):

• Total drag coefficient = (turbulent contribution over full length) - (correction for laminar starting length).
• The expression assumes Reynolds number at end of surface is much greater than critical value.

📐 Universal velocity profile

📐 Logarithmic law of the wall

The logarithmic velocity profile in the near-wall region is based on the premise of a constant shear layer very close to the wall.

Effective viscosity model:

• Total stress (viscous + turbulent) assumed constant, written in terms of friction velocity.
• Effective viscosity modeled as: (constant) × (friction velocity) × (distance from wall).
• The constant (kappa) is determined from experiments: κ ≈ 0.41.

Resulting velocity profile:

• Nondimensional velocity = (1/κ) × natural log of (nondimensional distance from wall) + B.
• B ≈ 5 from experimental results.
• Nondimensional distance = (friction velocity × distance from wall) / kinematic viscosity.

🎯 Regions of validity

Region	Nondimensional distance (y⁺)	Velocity profile	Notes
Viscous sublayer	y⁺ < 5	Linear: u⁺ = y⁺	Turbulence suppressed by wall; viscous forces dominate
Log-layer	30 < y⁺ < outer region	Logarithmic law	Valid over wide range of conditions (fluids, Reynolds numbers)
Outer region	Large y⁺ (increases with Re)	Deviates from log-law	Affected by pressure gradients along wall

Universal nature:

• The relationship of nondimensional velocity versus nondimensional distance has been shown to be very accurate over a wide range of conditions (different fluids, different Reynolds numbers, etc.).
• Valid fairly close to the surface in the constant stress layer.

Don't confuse: the log-profile is only valid in the intermediate region (30 < y⁺); very close to the wall (y⁺ < 5) the profile is linear, and far from the wall it transitions to the outer flow region.

🧭 Overview

🧠 One-sentence thesis

CFD replaces unsolvable analytical differential conservation equations with discrete algebraic approximations on a grid, enabling numerical solutions for complex flows across many disciplines despite inherent truncation errors and computational costs.

📌 Key points (3–5)

• What CFD does: discretizes conservation equations (mass, momentum, energy) over space and time grids to produce solvable algebraic systems instead of intractable analytical solutions.
• Why CFD is needed: analytical solutions exist only for the simplest flows; CFD enables simulation of complex geometries, transient effects, and multidisciplinary applications (weather, combustion, blood flow, etc.).
• Key trade-offs: smaller grid spacing and time steps reduce truncation error and improve accuracy, but exponentially increase computational cost and the number of unknowns.
• Common confusion—explicit vs implicit schemes: explicit methods solve one unknown per equation (fast per step, but require small time steps for stability via Courant number); implicit methods solve entire systems simultaneously (stable for larger time steps, but more complex per step).
• Turbulence modeling challenge: direct numerical simulation (DNS) resolves all fluctuations but is prohibitively expensive; Reynolds-averaged models (e.g., two-equation k-ε models) solve only mean flow by modeling turbulent stresses, trading exact detail for feasibility.

🔧 What CFD solves and why

🔧 The governing equations

CFD attempts to solve the basic (differential) conservation equations: conservation of mass, conservation of momentum (Navier-Stokes when stress is expressed via viscosity and strain rate), and conservation of energy for thermal problems.

• These are nonlinear partial differential equations for velocity and pressure (plus density for compressible flows, temperature for thermal problems).
• Analytical solutions are unavailable "for all but the most simple flows."
• The alternative: approximate solutions via discrete representations over grids in space and time → algebraic equation systems.

🌍 Why CFD matters

• Replaces expensive experiments: no physical model needed; parameters can be changed quickly to generate large data sets.
• Ubiquitous applications: geophysical flows, biomedical flows (arterial blood), combustion, weather prediction, pollution analysis, ocean mixing, avalanche prediction, etc.
• Specific capabilities: simulate complex geometries, moving surfaces (propellers, control flaps), transient injection processes, and calculate forces (pressure + wall shear stress integration), temperature fields, and heat flux distributions.
• Caution: results are not exact; sensitivity to discretization methods must be understood, and experimental verification is often essential.

🧮 How discretization works

🧮 The finite difference method

• Goal: replace derivatives with ratios of finite changes.
• Example: for a differential equation with spatial derivative, a Taylor series expansion around a grid point gives:
  • Forward difference: (u at x+Δx minus u at x) / Δx approximates the first derivative.
  • Truncation error is on the order of (Δx)² for this first-order scheme.
• Result: one algebraic equation per grid point; the system size equals the number of unknowns.
• Boundary conditions are inserted as known values at boundary grid points, reducing the number of unknowns.

📦 Finite volume method (alternative)

• Forms small "cells" or control volumes; conservation equations are written for each cell assuming single-valued variables at cell faces.
• Advantages over finite difference (not detailed in excerpt), but the net result is the same: a system of algebraic equations for unknowns at each cell.

📐 Grid and time discretization

• Space: domain divided into discrete points; a 3D grid with 1000 elements per coordinate = 10⁹ spatial points.
• Time: transient problems require discrete time steps; 1000 seconds of simulation → 10¹² data points per variable for the example above.
• Interpolation: values between grid points can be interpolated, introducing additional error.
• Example: the excerpt shows a multi-domain embedded grid around an oscillating airfoil, with progressively finer resolution near the surface.

⚖️ Accuracy, convergence, and stability

⚖️ Grid convergence

As grid size (Δx) decreases, the finite difference solution converges toward the exact analytical solution (when one exists).

• Truncation error decreases with smaller Δx, but the number of grid points (and computational cost) increases.
• The excerpt's example (a simple ODE with exact solution u = e^x) shows that error varies by location and shrinks as the number of discrete points N increases.
• Grid-independent solution: when further grid refinement produces negligible change, the solution is considered converged.

⚙️ Nonlinear effects and linearization

• If the governing equation includes a nonlinear term (e.g., u² when exponent m ≠ 1, analogous to the u·∇u term in Navier-Stokes), the discretized system becomes nonlinear.
• Iterative linearization: guess an initial value u*; express u² as u* · u; solve the linearized system for updated u; repeat until convergence.
• Residual: difference between solution and guess at each iteration; overall residual = sum over all grid points; convergence criterion is met when scaled residual falls below a threshold.

🛡️ Numerical stability

• Stable method: iterations converge toward a solution.
• Unstable method: solution diverges.
• Stability does not guarantee accuracy; a stable method can still have large errors if grid/time step is too coarse.
• For transient problems, time step size is critical; too large a step can cause divergence.

⏱️ Explicit vs implicit time-stepping

⏱️ Explicit schemes

• Time derivative discretized as (u at n+1 minus u at n) / Δt.
• All other terms evaluated at known time step n → only one unknown (u at n+1) per equation.
• Advantage: simple, fast per time step.
• Constraint: stability limited by the Courant number C = K · Δt / Δx (K has units of velocity); typically C < 1 for stability.
• Implication: as grid size Δx increases, allowable Δt can increase, but both large Δt and large Δx increase error.

⏱️ Implicit schemes

• Time derivative uses backward differencing: (u at n minus u at n-1) / Δt; all other terms at time step n.
• Multiple unknowns per equation → entire system must be solved simultaneously (implicit).
• Advantage: stable for much larger time steps than explicit schemes.
• Caution: larger time steps do not imply accuracy; numerical errors must still be controlled.

Scheme	Unknowns per equation	Stability constraint	Computational cost per step
Explicit	One (next time step)	Courant number < 1	Low
Implicit	Multiple (system-wide)	Much less restrictive	High (system solve)

🌀 Turbulence modeling

🌀 Why turbulence is hard

• Turbulent flows have fluctuations over a very wide range of time and length scales.
• Direct numerical simulation (DNS): resolve all scales with very small Δt and Δx → extremely computationally expensive, especially at high Reynolds numbers.
• Computing power is "nowhere near sufficient" for economically feasible DNS of reasonably large flows.

🌀 Reynolds averaging and turbulent stress

Reynolds averaging: decompose each variable into time-averaged mean (e.g., ū) plus fluctuating component (e.g., u' with time average zero).

• Time-averaging the nonlinear Navier-Stokes advection term produces the Reynolds stress: time average of products of fluctuating velocity components (e.g., ū'v').
• This term has units of stress per volume and represents the effect of turbulent fluctuations on the mean flow.
• Model: express Reynolds stress as turbulent viscosity × strain rate tensor − (2/3)·k·δ, where:
  • μ_t = turbulent viscosity (depends on the flow, not just the fluid).
  • k = turbulent kinetic energy (half the time average of squared fluctuating velocity).

🌀 Two-equation turbulence models (k-ε example)

• Turbulent kinematic viscosity modeled as ν_t ∝ k² / ε, where:
  • k = turbulent kinetic energy (energy per mass).
  • ε = turbulent dissipation rate (rate of loss of k; units: energy per time per mass).
• Dimensional analysis with empirical constant (0.09) gives ν_t = 0.09 · k² / ε.
• Two additional transport equations (derived from time-varying Navier-Stokes) are solved for k and ε at each grid point/cell.
• Result: mean velocity and pressure are solved from Reynolds-averaged Navier-Stokes equations using the modeled turbulent stress.
• Trade-off: significant added computational effort (two extra equations per grid location), but far less than DNS.

🌀 Practical considerations

• Multiple turbulence models available in commercial software; "best" choice depends on flow conditions and desired accuracy vs computational cost.
• Users must understand model limitations (typically described in software documentation).
• Solution yields time-averaged mean variables, not instantaneous fluctuations.

Don't confuse: DNS (resolves all fluctuations, prohibitively expensive) vs Reynolds-averaged models (solve only mean flow with modeled turbulent effects, feasible for practical problems).