Computational fluid dynamics

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11 Feb 2010 ... Computational Fluid Dynamics (CFD) is the discipline of discretizing these .... theorem. Note: Ω(t) is an ensemble of molecules, Ω is a fixed control volume. ...... integral (4)) have been developed, usually the latter class is used.
Computational fluid dynamics Martin Kronbichler [email protected] Applied Scientific Computing (Tillämpad beräkningsvetenskap)

February 11, 2010

Martin Kronbichler (TDB)

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Introduction The Navier–Stokes equations Conservation of mass Conservation of momentum Constitutive & kinematic relations

Conservation of energy Equations of state & compressible Navier–Stokes equations Incompressible Navier–Stokes equations Discretization Overview of spatial discretizations The Finite Volume Method Time discretization Spatial discretization

Turbulence and its modeling

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Introduction

What is CFD?

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Fluid mechanics deals with the motion of fluids (liquids and gases), induced by external forces.

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Fluid flow is modeled by partial differential equations (PDE), describing the conservation of mass, momentum, and energy.

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Computational Fluid Dynamics (CFD) is the discipline of discretizing these PDE and solving them using computers.

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Introduction

In which application fields is CFD used?

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Aerospace and aeronautical applications (airplanes, water and space vehicles)

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Mechanical applications (gas turbines, heat exchange, explosions, combustion, architecture)

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Biological applications (blood flow, breathing, drinking)

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Meteorological applications (weather prediction)

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Environmental applications (air and water pollution)

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and many more . . .

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Introduction

Why to use CFD?

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Introduction

Why to use CFD?

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Pre-design of components: simulation vs. experiment (simulations are cheaper, faster, and safer, but not always reliable) I I I I I

vehicles with lower fuel consumption, quieter, heavier loads combustion engines oil recovery water and gas turbines (effectiveness) stress minimization

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Introduction

Why to use CFD?

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Pre-design of components: simulation vs. experiment (simulations are cheaper, faster, and safer, but not always reliable) I I I I I

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vehicles with lower fuel consumption, quieter, heavier loads combustion engines oil recovery water and gas turbines (effectiveness) stress minimization

Detection and prediction I I I I

hurricanes, storms, tsunamis pollution transport diseases forces, stresses

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Introduction

Visualization of CFD results

ONERA M6 wing optimization O. Amoignon, M. Berggren

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Introduction

Visualization of CFD results

Pipe flow, computer lab

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Introduction

Requirements for industrial CFD

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Robustness — give a solution (for as many input cases as possible)

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Reliability — give a good solution

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Performance — give a good solution fast

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Geometries — give a good solution fast for real problems

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Automatic tool chain — reduce requirements of user interaction

Knowledgeable user controls and evaluates simulation outcomes!

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Introduction

CFD Solution Tool Chain I

CAD description of geometry

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Introduction

CFD Solution Tool Chain I

CAD description of geometry

I

Grid generation from CAD model (usually the most time consuming part)

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Introduction

CFD Solution Tool Chain I

CAD description of geometry

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Grid generation from CAD model (usually the most time consuming part) Problem set up - knowledgeable user needed

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I I I I

Mathematical model to use Spatial and temporal discretization Available resources Much more . . .

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Introduction

CFD Solution Tool Chain I

CAD description of geometry

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Grid generation from CAD model (usually the most time consuming part) Problem set up - knowledgeable user needed

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I I I I

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Mathematical model to use Spatial and temporal discretization Available resources Much more . . .

Preprocessing - parse configuration and prepare solver

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Introduction

CFD Solution Tool Chain I

CAD description of geometry

I

Grid generation from CAD model (usually the most time consuming part) Problem set up - knowledgeable user needed

I

I I I I

Mathematical model to use Spatial and temporal discretization Available resources Much more . . .

I

Preprocessing - parse configuration and prepare solver

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Solving - run the solver

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Introduction

CFD Solution Tool Chain I

CAD description of geometry

I

Grid generation from CAD model (usually the most time consuming part) Problem set up - knowledgeable user needed

I

I I I I

Mathematical model to use Spatial and temporal discretization Available resources Much more . . .

I

Preprocessing - parse configuration and prepare solver

I

Solving - run the solver

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Post processing - extract and compute information of interest

Martin Kronbichler (TDB)

FVM for CFD

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Introduction

CFD Solution Tool Chain I

CAD description of geometry

I

Grid generation from CAD model (usually the most time consuming part) Problem set up - knowledgeable user needed

I

I I I I

Mathematical model to use Spatial and temporal discretization Available resources Much more . . .

I

Preprocessing - parse configuration and prepare solver

I

Solving - run the solver

I

Post processing - extract and compute information of interest

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Visualization - often most important

Martin Kronbichler (TDB)

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Introduction

CFD Solution Tool Chain I

CAD description of geometry

I

Grid generation from CAD model (usually the most time consuming part) Problem set up - knowledgeable user needed

I

I I I I

Mathematical model to use Spatial and temporal discretization Available resources Much more . . .

I

Preprocessing - parse configuration and prepare solver

I

Solving - run the solver

I

Post processing - extract and compute information of interest

I

Visualization - often most important

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Interpretation - physics, mathematics, numerics, experiments

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Introduction

CFD software

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Commercial: Fluent, Comsol, CFX, Star-CD

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In-house codes: Edge (FOI), DLR-Tau (German Aerospace Center), Fun3D (NASA), Sierra/Premo (American Aerospace)

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Open Source: OpenFOAM, FEniCS, OpenFlower

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Introduction

CFD links

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http://www.cfd-online.com

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http://www.fluent.com

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http://www.openfoam.org

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The Navier–Stokes equations

A mathematical model for fluid flow

The Navier–Stokes equations

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The Navier–Stokes equations

Navier–Stokes equations — an overview

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The governing equations of fluid dynamics are the conservation laws of mass, momentum, and energy.

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In CFD, this set of conservation laws are called the Navier–Stokes equations.

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Derive compressible Navier–Stokes equations first

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Simplify these equations to get the incompressible Navier–Stokes equations

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The Navier–Stokes equations

Notation

density velocity, u = (u1 , u2 , u3 ) pressure temperature dynamic viscosity kinematic viscosity,  ν = µ/ρ

ρ u p T µ ν ∇ ·

nabla operator, ∇ =

∂ ∂ ∂ ∂x1 , ∂x2 , ∂x3



inner product, a · b = a1 b1 + a2 b2 + a3 b3

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The Navier–Stokes equations

General form of conservation laws I I

Monitor the flow characteristics of a fixed control volume Ω. Rate of total change in the control volume: I I

change in the interior, flow over boundary of control volume,

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The Navier–Stokes equations

General form of conservation laws I I

Monitor the flow characteristics of a fixed control volume Ω. Rate of total change in the control volume: I I

change in the interior, flow over boundary of control volume,

expressed as Z Z Z Z ∂q D q dΩ = dΩ + qu · n ds = S dΩ Dt Ω(t) Ω ∂t ∂Ω Ω(t) for some quantity q and a source term S (generation or elimination). n denotes the outer normal on Ω. This is the Reynolds transport theorem. Note: Ω(t) is an ensemble of molecules, Ω is a fixed control volume.

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The Navier–Stokes equations

General form of conservation laws I I

Monitor the flow characteristics of a fixed control volume Ω. Rate of total change in the control volume: I I

change in the interior, flow over boundary of control volume,

expressed as Z Z Z Z ∂q D q dΩ = dΩ + qu · n ds = S dΩ Dt Ω(t) Ω ∂t ∂Ω Ω(t) for some quantity q and a source term S (generation or elimination). n denotes the outer normal on Ω. This is the Reynolds transport theorem. Note: Ω(t) is an ensemble of molecules, Ω is a fixed control volume. I

Differential form:

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The Navier–Stokes equations

Conservation of mass

Continuity equation I

The continuity equation describes the conservation of mass . The mass M of the material in Ω(t) is constant (molecules cannot be created/destroyed), that is M(t) = M(t + ∆t). Z DM D = ρ dΩ = 0 Dt Dt Ω(t) | {z } M ρ(t) denotes the fluid density.

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The Navier–Stokes equations

Conservation of mass

Continuity equation II

Rewritten for a stationary control volume by using the Reynolds transport theorem: rate of mass change in Ω + mass flow over ∂Ω = 0, Z Z ∂ρ dΩ + ρu · n ds = 0. Ω ∂t ∂Ω

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The Navier–Stokes equations

Conservation of momentum

Momentum equation I

The momentum equation describes the conservation of momentum . Newton’s second law of motion: the total rate of momentum m change in Ω(t) is equal to the sum of acting forces K, i.e., Z Dm D = ρu d Ω = K, Dt Dt Ω(t) (interpretation: mass × acceleration = force).

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The Navier–Stokes equations

Conservation of momentum

Momentum equation II Rewritten for a stationary control volume Ω: Z Z ∂ρu + ρ(u ⊗ u) · n ds = K, Ω ∂t ∂Ω where u ⊗ u is a rank-2 tensor with entries ui uj .

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The Navier–Stokes equations

Conservation of momentum

Momentum equation II Rewritten for a stationary control volume Ω: Z Z ∂ρu + ρ(u ⊗ u) · n ds = K, Ω ∂t ∂Ω where u ⊗ u is a rank-2 tensor with entries ui uj . Decompose force into surface forces and volume forces: rate of momentum change in Ω + momentum flow over ∂Ω = surface forces on ∂Ω + volume forces on Ω , Z Z Z Z Z ∂ρu dV + ρ(u⊗u)·n ds = − pn ds + τ · n ds + ρf d Ω. Ω ∂t ∂Ω Ω | ∂Ω {z ∂Ω } surface forces

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The Navier–Stokes equations

Conservation of momentum

Momentum equation III

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Surface forces acting on ds of ∂Ω: I

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Pressure force p(−n) ds (analog to isotropic stress in structural mechanics), Viscous force τ · n ds, τ is the Cauchy stress tensor (analog to deviatoric stress in structural mechanics).

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The Navier–Stokes equations

Conservation of momentum

Momentum equation III

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Surface forces acting on ds of ∂Ω: I

I

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Pressure force p(−n) ds (analog to isotropic stress in structural mechanics), Viscous force τ · n ds, τ is the Cauchy stress tensor (analog to deviatoric stress in structural mechanics).

Volume forces ρf d Ω acting on small interior volume d Ω: I I

Gravity force ρg d Ω, e.g., f = g, Other types of forces: Coriolis, centrifugal, electromagnetic, buoyancy

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The Navier–Stokes equations

Conservation of momentum

Newtonian fluid Model that relates the stress tensor τ to the velocity u: We consider so-called Newtonian fluids, where the viscous stress is linearly related to strain rate, that is,   2 2 τ = 2µε(u) − µI tr ε(u) = µ ∇u + (∇u)T − µ(∇ · u)I, 3 3 (compare with constitutive and kinematic relations in elasticity theory).

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The Navier–Stokes equations

Conservation of momentum

Newtonian fluid Model that relates the stress tensor τ to the velocity u: We consider so-called Newtonian fluids, where the viscous stress is linearly related to strain rate, that is,   2 2 τ = 2µε(u) − µI tr ε(u) = µ ∇u + (∇u)T − µ(∇ · u)I, 3 3 (compare with constitutive and kinematic relations in elasticity theory). For Cartesian coordinates  τij = µ

∂uj ∂ui + ∂xi ∂xj



3

2 X ∂uk . − µ 3 ∂xk k=1

The dynamic viscosity is assumed constant µ = µ(T , p) ≈ constant. Martin Kronbichler (TDB)

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The Navier–Stokes equations

Conservation of energy

Energy equation I

The energy equation describes the conservation of energy . The first law of thermodynamics: The total rate of total energy E changes in Ω(t) is equal to the rate of work L done on the fluid by the acting forces K plus the rate of heat added W , that is Z DE D = ρE d Ω = L + W Dt Dt Ω(t) where ρE is the total energy per unit volume.

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The Navier–Stokes equations

Conservation of energy

Energy Equation II Rewritten for a stationary control volume Ω, explicitly specifying the source terms: Rate of total energy change in Ω + Total energy flow over ∂Ω = Rate of work of pressure and viscous forces on ∂Ω + Rate of work of forces on Ω + Rate of heat added over ∂Ω Z Z Z Z ∂ρE dΩ + ρE u · n ds = − pu · n ds + (τ · u) · n ds ∂Ω Ω ∂t Z∂Ω Z ∂Ω + ρf · u d Ω + k∇T · n ds Ω

∂Ω

for the fluid temperature T and thermal conductivity k.

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The Navier–Stokes equations

Conservation of energy

Involved variables and equations

We have I

seven variables (ρ, u1 , u2 , u3 , p, T , E )

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five equations (continuity, 3×momentum, energy)

We need two more equations to close the system, the so-called equations of state.

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The Navier–Stokes equations

Equations of state & compressible Navier–Stokes equations

Equations of state Properties of a perfect gas: internal energy

z }| { 1 2 p =(γ − 1)ρ E − |u| 2   1 1 T = E − |u|2 cv 2 c

where γ = cvp is the ratio of specific heats, and cp and cv are the specific heats at constant pressure and volume. A typical value for air at sea level pressure is γ = 1.4.

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The Navier–Stokes equations

Equations of state & compressible Navier–Stokes equations

The compressible Navier–Stokes equations in integral form The equations of continuity, momentum, and energy can be combined into one system of equations. I

I

Define compound variable U = (ρ, ρu1 , ρu2 , ρu3 , ρE ) (called conservative variables) Define flux vectors     ρu · n 0  τ ·n F · n =  ρ(u ⊗ u) · n + pI · n  −  (ρE + p)u · n (τ · u) · n + k(∇T ) · n | {z } | {z } inviscid/convective

viscous

and an external strength vector Fe = (0, f, f · u)T

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The Navier–Stokes equations

Equations of state & compressible Navier–Stokes equations

The compressible Navier–Stokes equations in integral form The equations of continuity, momentum, and energy can be combined into one system of equations. I

I

Define compound variable U = (ρ, ρu1 , ρu2 , ρu3 , ρE ) (called conservative variables) Define flux vectors     ρu · n 0  τ ·n F · n =  ρ(u ⊗ u) · n + pI · n  −  (ρE + p)u · n (τ · u) · n + k(∇T ) · n | {z } | {z } inviscid/convective

viscous

and an external strength vector Fe = (0, f, f · u)T Compressible Navier–Stokes equations in integral form Z Z Z ∂U F · n ds = ρFe d Ω dΩ + Ω ∂t ∂Ω Ω Martin Kronbichler (TDB)

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The Navier–Stokes equations

Equations of state & compressible Navier–Stokes equations

Compressible Navier–Stokes equations, differential form Assume: flux tensor F is differentiable Apply the Gauss theorem to the integral form and get  Z  ∂U + ∇ · F − ρFe d Ω = 0. ∂t Ω

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The Navier–Stokes equations

Equations of state & compressible Navier–Stokes equations

Compressible Navier–Stokes equations, differential form Assume: flux tensor F is differentiable Apply the Gauss theorem to the integral form and get  Z  ∂U + ∇ · F − ρFe d Ω = 0. ∂t Ω Since the integral is zero for an arbitrary control volume Ω, we obtain the differential form of the Compressible Navier–Stokes equations ∂U + ∇ · F = ρFe . ∂t

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The Navier–Stokes equations

Incompressible Navier–Stokes equations

The incompressible Navier–Stokes equations

Motivation: we can use simpler equations if we have more information I

Incompressible fluid: density does not change with pressure, i.e., ρ = const.

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Energy equation decouples from the rest of the system; continuity and momentum equations can be simplified.

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Many practically relevant flows are incompressible (e.g. air at speeds up to 100 m/s) — predominant CFD model

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The Navier–Stokes equations

Incompressible Navier–Stokes equations

Continuity equation

The continuity equation, ∂ρ + ∇ · (ρu) = 0, ∂t becomes for constant ρ simply ∇ · u = 0,

i.e.,

3 X ∂ui i=1

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∂xi

= 0.

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The Navier–Stokes equations

Incompressible Navier–Stokes equations

Momentum equation The momentum equation, ∂ρu + ∇ · (ρu ⊗ u) = −∇p + ∇ · τ + ρf, ∂t can be simplified by using ∇ · u = 0, ∇ · (u ⊗ u) = (u · ∇)u + (∇ · u)u = (u · ∇)u   2   τ = µ ∇u + (∇u)T − µ(∇ · u)I = ∇u + (∇u)T 3

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The Navier–Stokes equations

Incompressible Navier–Stokes equations

Momentum equation The momentum equation, ∂ρu + ∇ · (ρu ⊗ u) = −∇p + ∇ · τ + ρf, ∂t can be simplified by using ∇ · u = 0, ∇ · (u ⊗ u) = (u · ∇)u + (∇ · u)u = (u · ∇)u   2   τ = µ ∇u + (∇u)T − µ(∇ · u)I = ∇u + (∇u)T  3  T = µ ∇2 u + ∇(∇ · u) = µ∇2 u. ∇ · τ = µ∇ · ∇u + (∇u)

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The Navier–Stokes equations

Incompressible Navier–Stokes equations

Momentum equation The momentum equation, ∂ρu + ∇ · (ρu ⊗ u) = −∇p + ∇ · τ + ρf, ∂t can be simplified by using ∇ · u = 0, ∇ · (u ⊗ u) = (u · ∇)u + (∇ · u)u = (u · ∇)u   2   τ = µ ∇u + (∇u)T − µ(∇ · u)I = ∇u + (∇u)T  3  T = µ ∇2 u + ∇(∇ · u) = µ∇2 u. ∇ · τ = µ∇ · ∇u + (∇u) This gives the incompressible version of the momentum equation ∂u 1 µ + (u · ∇)u = − ∇p + ∇2 u + f ∂t ρ ρ Martin Kronbichler (TDB)

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The Navier–Stokes equations

Incompressible Navier–Stokes equations

The incompressible Navier–Stokes equations II

Since the energy equation does not enter the momentum and continuity equation, we have a closed system of four equations: Incompressible Navier–Stokes equations   ∇ · u = 0, 1 ∂u  + (u · ∇)u = − ∇p + ν∇2 u + f, ∂t ρ where ν =

µ ρ

is the fluid kinematic viscosity.

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The Navier–Stokes equations

Incompressible Navier–Stokes equations

Equation in temperature The energy equation can be expressed in terms of temperature by using the equation of state. Transforming the integral form into differential form as done before1 yields   ∂T ρcp + ∇ · (T u) = ∇ · (k∇T ) + τ : ∇u, ∂t where τ : ∇u =

3 X i,j=1

  3 X ∂uj 2 ∂uj 1 ∂ui =µ + , τij ∂xi 2 ∂xj ∂xi i,j=1

and is called a dissipative function.

1 for further reading, see e.g. J. Blazek: Computational Fluid Dynamics, Elsevier, Amsterdam Martin Kronbichler (TDB)

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The Navier–Stokes equations

Incompressible Navier–Stokes equations

Incompressible Navier–Stokes equations III I

Energy equation is decoupled from the continuity and momentum equations ⇒ I

I

first solve the continuity and momentum equations to get the velocity and pressure. temperature (energy) equation can be solved for temperature using the already computed velocity.

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The Navier–Stokes equations

Incompressible Navier–Stokes equations

Incompressible Navier–Stokes equations III I

Energy equation is decoupled from the continuity and momentum equations ⇒ I

I

I

first solve the continuity and momentum equations to get the velocity and pressure. temperature (energy) equation can be solved for temperature using the already computed velocity.

Pressure level only defined up to a constant for incompressible flow ⇒ I I

pressure level has to be fixed at one point in the flow. p yields then the relative pressure difference with respect to that pressure level.

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The Navier–Stokes equations

Incompressible Navier–Stokes equations

Incompressible Navier–Stokes equations III I

Energy equation is decoupled from the continuity and momentum equations ⇒ I

I

I

Pressure level only defined up to a constant for incompressible flow ⇒ I I

I

first solve the continuity and momentum equations to get the velocity and pressure. temperature (energy) equation can be solved for temperature using the already computed velocity.

pressure level has to be fixed at one point in the flow. p yields then the relative pressure difference with respect to that pressure level.

No time derivative for the pressure ⇒ I I

mathematical difficulty of the incompressible Navier–Stokes equations. p is indirectly determined by the condition ∇ · u = 0 (p has to be such that the resulting velocity is divergence free, Lagrange multiplier).

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The Navier–Stokes equations

Incompressible Navier–Stokes equations

Additional conditions

To complete the physical problem setup after having derived the appropriate PDE, we have to define the I

domain

I

initial conditions (velocity, temperature, etc at t = 0)

I

boundary conditions (inflow, outflow, wall, interface, . . .)

I

material properties

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Discretization

Discretization

Numerical solution strategies for the flow equations

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Discretization

Solution Strategies Outline

Some points we will cover regarding solver strategies of CFD I

Pros and cons of various discretization methods

I

Finite Volume Method (FVM)

I

Turbulence and its modeling

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Discretization

Overview of spatial discretizations

Overview of Methods Discretization methods for the CFD equations: I

Finite Difference Method (FDM) + efficiency, + theory, − geometries

I

Finite Element Method (FEM) + theory, + geometries, − shocks, − uses no directional information

I

Spectral Methods (Collocation, Galerkin, . . .) + accuracy, + theory, − geometries

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Discretization

Overview of spatial discretizations

Overview of Methods Discretization methods for the CFD equations: I

Finite Difference Method (FDM) + efficiency, + theory, − geometries

I

Finite Element Method (FEM) + theory, + geometries, − shocks, − uses no directional information

I

Spectral Methods (Collocation, Galerkin, . . .) + accuracy, + theory, − geometries

I

Finite Volume Methods (FVM) + robustness, + geometries, − accuracy

I

Discontinuous Galerkin Methods (DGM) + geometries, + shocks, − efficiency (e.g. smooth solutions)

Martin Kronbichler (TDB)

FVM for CFD

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Discretization

Overview of spatial discretizations

Overview of Methods Discretization methods for the CFD equations: I

Finite Difference Method (FDM) + efficiency, + theory, − geometries

I

Finite Element Method (FEM) + theory, + geometries, − shocks, − uses no directional information

I

Spectral Methods (Collocation, Galerkin, . . .) + accuracy, + theory, − geometries

I

Finite Volume Methods (FVM) + robustness, + geometries, − accuracy

I

Discontinuous Galerkin Methods (DGM) + geometries, + shocks, − efficiency (e.g. smooth solutions)

I

Hybrid Methods + versatile, − complex (not automatable)

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Discretization

The Finite Volume Method

Introduction

The Finite Volume Method (FVM) is based on the integral form of the governing equations. The integral conservation is enforced in so-called control volumes ( d Ω → ∆Ω) defined by the computational mesh. The type of FVM is specified by I

the type of control volume

I

the type of evaluation of integrals and fluxes

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Discretization

The Finite Volume Method

FVM derivation, introduction Consider a general scalar hyperbolic conservation law: ∂u + ∇ · F(u) = 0 in Ω, ∂t

(1)

with appropriate boundary and initial conditions.

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Discretization

The Finite Volume Method

FVM derivation, introduction Consider a general scalar hyperbolic conservation law: ∂u + ∇ · F(u) = 0 in Ω, ∂t

(1)

with appropriate boundary and initial conditions. Compared to the compressible Navier–Stokes equations, we use a problem where I

u replaces the vector of conservative variables (ρ, ρu, ρE ),

I

F replaces the flux tensor as defined earlier,

I

we have a scalar problem instead of a system.

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Discretization

The Finite Volume Method

FVM derivation, control volumes Consider the equation in two dimensions, and we divide the domain Ω into M non-overlapping control volumes Km ⊂ Ω.

Km

vertex-centered finite volume method Martin Kronbichler (TDB)

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Discretization

The Finite Volume Method

FVM derivation, discretized solution

In each control volume, Km , we store one value um , which is the average of u in Km . Z 1 um = u dΩ |Km | Km

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Discretization

The Finite Volume Method

FVM derivation, integral form I The FVM is based on the integral formulation of the equations. We start by integrating (1) over one of the control volumes, Z Z ∂u dΩ + ∇ · F(u) d Ω = 0. Km ∂t Km

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February 11, 2010

(2)

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Discretization

The Finite Volume Method

FVM derivation, integral form I The FVM is based on the integral formulation of the equations. We start by integrating (1) over one of the control volumes, Z Z ∂u dΩ + ∇ · F(u) d Ω = 0. Km ∂t Km

(2)

The first integral can be simplified by changing order of time derivative and integration to get Z Z ∂u d dum dΩ = u d Ω = |Km | . (3) ∂t dt dt Km Km Inserting (3) back into (2) yields dum 1 =− dt |Km | Martin Kronbichler (TDB)

Z ∇ · F(u) d Ω. Km

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Discretization

The Finite Volume Method

FVM derivation, time evolution We need to propagate the solution um in time. There are many possibilities for choosing a time integration method, for example Runge–Kutta, multistep methods etc. See some Ordinary Differential Equation (ODE) textbook for more information.2

2 e.g., E. Hairer, S.P. Nørsett, G. Wanner: Solving Ordinary Differential Equations. I: Nonstiff Problems. Springer-Verlag, Berlin, 1993. Martin Kronbichler (TDB)

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Discretization

The Finite Volume Method

FVM derivation, time evolution We need to propagate the solution um in time. There are many possibilities for choosing a time integration method, for example Runge–Kutta, multistep methods etc. See some Ordinary Differential Equation (ODE) textbook for more information.2 We will use forward Euler in all our examples, i.e., n dum u n+1 − um = m , dt ∆t n = u (t ). where um m n

2 e.g., E. Hairer, S.P. Nørsett, G. Wanner: Solving Ordinary Differential Equations. I: Nonstiff Problems. Springer-Verlag, Berlin, 1993. Martin Kronbichler (TDB)

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Discretization

The Finite Volume Method

FVM derivation, time evolution

Inserting the forward Euler time discretization into our system gives Z ∆t n+1 n um = um − ∇ · F(u) d Ω. |Km | Km

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Discretization

The Finite Volume Method

FVM derivation, integral form II By integration by parts (Gauss divergence theorem) of the flux integral we get the basic FVM formulation Z ∆t n+1 n um = um − F(u) · n ds. (4) |Km | ∂Km The boundary integral describes the flux of u over the boundary ∂Km of the control volume. Note: Transforming the volume to a surface integral gets us back to the form used for the derivation of the Navier–Stokes equations.

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Discretization

The Finite Volume Method

FVM derivation, integral form II By integration by parts (Gauss divergence theorem) of the flux integral we get the basic FVM formulation Z ∆t n+1 n um = um − F(u) · n ds. (4) |Km | ∂Km The boundary integral describes the flux of u over the boundary ∂Km of the control volume. Note: Transforming the volume to a surface integral gets us back to the form used for the derivation of the Navier–Stokes equations. Different ways of evaluating the flux integral specify the type of FVM (together with the control volume type).

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Discretization

The Finite Volume Method

FVM derivation, numerical flux

The integral in (4) is evaluated using a so-called numerical flux function. I

3

There are a lot of different numerical fluxes available for the integral evaluation, each of them having its pros and cons. Terminology: Riemann solvers (they are usually designed to solve the Riemann problem which is a conservation law with constant data and a discontinuity).

Contains only the inviscid terms from the compressible Navier–Stokes equations

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Discretization

The Finite Volume Method

FVM derivation, numerical flux

The integral in (4) is evaluated using a so-called numerical flux function. I

I

There are a lot of different numerical fluxes available for the integral evaluation, each of them having its pros and cons. Terminology: Riemann solvers (they are usually designed to solve the Riemann problem which is a conservation law with constant data and a discontinuity). Both exact and approximate solvers (i.e., ways to evaluate the flux integral (4)) have been developed, usually the latter class is used. I I

3

E.g. exact solver for Euler3 equations developed by Godunov. Widely used approximate solvers: Roe, HLLC, central flux.

Contains only the inviscid terms from the compressible Navier–Stokes equations

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Discretization

The Finite Volume Method

FVM derivation, numerical flux II The numerical flux functions are usually denoted by F ∗ (uL , uR , n) where uL is the left (local) state and uR is the right (remote) state of u at the boundary ∂Km , Z XZ F(u) · n ds = F(u) · nj ds ∂Km

Martin Kronbichler (TDB)

j

j ∂Km

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Discretization

The Finite Volume Method

FVM derivation, numerical flux II The numerical flux functions are usually denoted by F ∗ (uL , uR , n) where uL is the left (local) state and uR is the right (remote) state of u at the boundary ∂Km , Z XZ F(u) · n ds = F(u) · nj ds ∂Km

j



X

j ∂Km

|∂Kmj |F ∗ (uL , uR , nj )

j

and j denotes the index of each subboundary of ∂Km with a given normal nj .

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Discretization

The Finite Volume Method

FVM derivation, full discretization The full discretization for control volume Km is hence n+1 n um = um −

∆t X |∂Kmj |F ∗ (uL , uR , nj ) |Km | j

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Discretization

The Finite Volume Method

FVM derivation, full discretization The full discretization for control volume Km is hence n+1 n um = um −

∆t X |∂Kmj |F ∗ (uL , uR , nj ) |Km | j

In one dimension, we have n+1 um

Z ∆t = − F(u) · n ds ∆x ∂Km  ∆t  n n n = um − F(um+1/2 ) − F(um−1/2 ) ∆x n um

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Discretization

The Finite Volume Method

FVM derivation, full discretization The full discretization for control volume Km is hence n+1 n um = um −

∆t X |∂Kmj |F ∗ (uL , uR , nj ) |Km | j

In one dimension, we have n+1 um

Z ∆t = − F(u) · n ds ∆x ∂Km  ∆t  n n n = um − F(um+1/2 ) − F(um−1/2 ) ∆x  ∆t n n n n n = um − F ∗ (um , um−1 , −1) + F ∗ (um , um+1 , 1) ∆x n um

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Discretization

The Finite Volume Method

FVM derivation, upwind scheme I Get the information upwind, where the information comes from.

Image from http://en.wikipedia.org

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Discretization

The Finite Volume Method

FVM derivation, upwind scheme II Example in one dimension F(u) = βu, where β is the direction of the flow. So in each control volume we compute Z βun ds. ∂Km

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Discretization

The Finite Volume Method

FVM derivation, upwind scheme II Example in one dimension F(u) = βu, where β is the direction of the flow. So in each control volume we compute Z βun ds. ∂Km

Case β > 0 (information flows from left to right), equidistant mesh: um−1/2 = um−1 ,

Martin Kronbichler (TDB)

um+1/2 = um .

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Discretization

The Finite Volume Method

FVM derivation, upwind scheme II Example in one dimension F(u) = βu, where β is the direction of the flow. So in each control volume we compute Z βun ds. ∂Km

Case β > 0 (information flows from left to right), equidistant mesh: um−1/2 = um−1 ,

um+1/2 = um .

Hence, we get  ∆t  n n F(um+1/2 )nm+1/2 + F(um−1/2 )nm−1/2 ∆x   ∆t  n ∆t n n n n n = um − β um+1/2 − um−1/2 = um − β um − um−1 . ∆x ∆x

n+1 n um = um −

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Discretization

The Finite Volume Method

FVM derivation, upwind scheme (Roe flux)

Practical implementation of upwinding: construct it by the Roe Flux F ∗ (uL , uR , n) =

uL − uR n · F(uL ) + n · F(uR ) + n · F 0 (¯ u ) . 2 2

Here, u¯ satisfies the mean value theorem, n · F(uL ) = n · F(uR ) + n · F 0 (¯ u ) (uL − uR ) . and is called the Roe average (problem dependent).

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Discretization

The Finite Volume Method

FVM derivation, upwind scheme (Roe flux)

Practical implementation of upwinding: construct it by the Roe Flux F ∗ (uL , uR , n) =

uL − uR n · F(uL ) + n · F(uR ) + n · F 0 (¯ u ) . 2 2

Here, u¯ satisfies the mean value theorem, n · F(uL ) = n · F(uR ) + n · F 0 (¯ u ) (uL − uR ) . and is called the Roe average (problem dependent). Upwinding is first order accurate.

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Discretization

The Finite Volume Method

FVM derivation, central scheme I For F(u) = βu on a equidistant mesh, the central scheme approximation is um−1/2 =

Martin Kronbichler (TDB)

um−1 + um , 2

um−1/2 =

FVM for CFD

um + um+1 . 2

February 11, 2010

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Discretization

The Finite Volume Method

FVM derivation, central scheme I For F(u) = βu on a equidistant mesh, the central scheme approximation is um−1/2 =

um−1 + um , 2

um−1/2 =

um + um+1 . 2

Hence,  ∆t  n n F(um+1/2 )nm+1/2 + F(um−1/2 )nm−1/2 ∆x  ∆t  n n n β um+1/2 − um−1/2 − = um ∆x  n n  n + un um um−1 + um ∆t m+1 n = um − β − ∆x 2 2  n  n um+1 − um−1 ∆t n = um − β . ∆x 2

n+1 n − um = um

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Discretization

The Finite Volume Method

FVM derivation, central scheme II Possibilities for evaluating the central flux:   uR − d , 1. F ∗ (uL , uR , n) = n · F uL + 2 2. F ∗ (uL , uR , n) =

Martin Kronbichler (TDB)

n · F(uL ) + n · F(uR ) − d. 2

FVM for CFD

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Discretization

The Finite Volume Method

FVM derivation, central scheme II Possibilities for evaluating the central flux:   uR − d , 1. F ∗ (uL , uR , n) = n · F uL + 2 2. F ∗ (uL , uR , n) =

n · F(uL ) + n · F(uR ) − d. 2

The central scheme is second order accurate. But: I

the central scheme gives rise to unphysical oscillations around steep gradients (shocks)

I

the central scheme ignores the direction of the flow

I

choice of artificial dissipation d can reduce accuracy

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Discretization

The Finite Volume Method

FVM derivation, second derivatives

Flux tensor F contains terms with first derivatives in u for the Navier–Stokes equations (corresponding to second derivatives in the differential form of the equations). Need to approximate these terms before evaluating the flux function.

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Discretization

The Finite Volume Method

FVM derivation, second derivatives

Flux tensor F contains terms with first derivatives in u for the Navier–Stokes equations (corresponding to second derivatives in the differential form of the equations). Need to approximate these terms before evaluating the flux function. Usually, apply central differences of the kind   ∂u um − um−1 . = ∂x m−1/2 ∆x

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Discretization

The Finite Volume Method

Higher order discretization schemes for FVM There are a number of higher order schemes for orders of accuracy ≥ 2, like I

MUSCL (Monotone Upstream-centered Schemes for Conservation Laws)

I

ENO (Essentially Non-Oscillatory)

I

WENO (Weighted ENO)

I

RDS (Residual Distribution Scheme)

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Discretization

The Finite Volume Method

Higher order discretization schemes for FVM There are a number of higher order schemes for orders of accuracy ≥ 2, like I

MUSCL (Monotone Upstream-centered Schemes for Conservation Laws)

I

ENO (Essentially Non-Oscillatory)

I

WENO (Weighted ENO)

I

RDS (Residual Distribution Scheme)

They combine different techniques to attain a high-order solution without excessive oscillations, for example so called flux limiters (reducing artificial oscillations), wider stencils over several control volumes, different weightings, reconstruction, and adding degrees of freedom in each control volume.

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FVM for CFD

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Discretization

The Finite Volume Method

Choosing the discretization parameters A necessary condition for stability of the time discretization is the CFL condition (depends on the combination of space and time discretization). For the one dimensional upwind discretization n+1 n um = um −

 ∆t n n β um − um−1 , ∆x

we require ∆t ∆x β ≤ 1. Interpretation: The difference approximation (which only uses information from one grid point to the left/right) can only represent variations in the solution up to β/∆x from one time step to the next → time step size has to be smaller than that limit

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Discretization

The Finite Volume Method

Steady state calculations For certain computations, the time-dependance is of no importance. Sought: steady state solution. when the solution has stabilized, the change in time will go to zero, n+1 n um = um −

∆t X n |∂Kmj |F ∗ (uL , uR , nj ) = um , |Km | j | {z }

∀m.

residual

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Discretization

The Finite Volume Method

Steady state calculations For certain computations, the time-dependance is of no importance. Sought: steady state solution. when the solution has stabilized, the change in time will go to zero, n+1 n um = um −

∆t X n |∂Kmj |F ∗ (uL , uR , nj ) = um , |Km | j | {z }

∀m.

residual

Measure whether steady state has been achieved: residual gets small. To reach steady state, many different acceleration techniques can be used, for example local time stepping and multigrid.

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February 11, 2010

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Discretization

The Finite Volume Method

Steady state calculations For certain computations, the time-dependance is of no importance. Sought: steady state solution. when the solution has stabilized, the change in time will go to zero, n+1 n um = um −

∆t X n |∂Kmj |F ∗ (uL , uR , nj ) = um , |Km | j | {z }

∀m.

residual

Measure whether steady state has been achieved: residual gets small. To reach steady state, many different acceleration techniques can be used, for example local time stepping and multigrid. Two examples of steady state: I

computation of the drag and lift around an airfoil,

I

mixing problem considered in the computer lab.

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Discretization

The Finite Volume Method

FVM for the incompressible Navier–Stokes equations

I

The incompressible Navier–Stokes equations are a system in five equations (u, p, T ), so go through them one by one according to the above procedure

I

Need to take special care of p, since it does not involve a time derivative

I

Computational domain Ω and material parameters ρ, ν specified from application

I

Initial and boundary conditions complete the formulation

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Discretization

The Finite Volume Method

Initial condition

The state of all variables at time t0 has to be defined in order to initiate the solution process. For example u|t=0 = u0 .

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Discretization

The Finite Volume Method

Boundary conditions, inflow The inflow is the part of the domain ∂Ω where u · n < 0.

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Discretization

The Finite Volume Method

Boundary conditions, inflow The inflow is the part of the domain ∂Ω where u · n < 0. Setting an inflow Dirichlet boundary condition, that is u = uin on ∂Kin : I

strongly imposed, set the inflow boundary node values to the respective value Um = Uin

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for all faces ∂Km on ∂Kin .

FVM for CFD

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Discretization

The Finite Volume Method

Boundary conditions, inflow The inflow is the part of the domain ∂Ω where u · n < 0. Setting an inflow Dirichlet boundary condition, that is u = uin on ∂Kin : I

strongly imposed, set the inflow boundary node values to the respective value Um = Uin

I

for all faces ∂Km on ∂Kin .

weakly imposed, use the numerical flux, F ∗ (Um , Uin , n). The inflow data can be viewed as given in a ghost point; boundary treated as the interior.

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Discretization

The Finite Volume Method

Boundary conditions, outflow The outflow is the part of the domain ∂Ω where u · n > 0. Treating outflow boundaries is more complicated than inflow parts. Usually one does a weak formulation such that 1. F ∗ (UL , UL , n) is used as flux function, 2. F ∗ (UL , U∞ , n) is used as flux function with U∞ a far-field velocity/pressure/temperature. There are also strong formulations.

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Discretization

The Finite Volume Method

Boundary conditions, outflow The outflow is the part of the domain ∂Ω where u · n > 0. Treating outflow boundaries is more complicated than inflow parts. Usually one does a weak formulation such that 1. F ∗ (UL , UL , n) is used as flux function, 2. F ∗ (UL , U∞ , n) is used as flux function with U∞ a far-field velocity/pressure/temperature. There are also strong formulations. Beware of artificial back flows! A common trick is to make the domain big enough to avoid distorting the solution in the domain of interest.

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Discretization

The Finite Volume Method

Boundary conditions, solid walls

The solid wall is the part of the domain ∂Ω where u · n = 0. For viscous (Navier–Stokes) flow we apply a no-slip condition u = 0. These boundary conditions are usually imposed strongly.

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Discretization

The Finite Volume Method

Heat transfer

When considering heat transfer for the incompressible Navier–Stokes equations, we also need to assign a BC for the temperature T . Two mechanisms can be applied: I

specify temperature with a Dirichlet BC, T = Tw , or

I

specify heat flux with a Neumann BC, ∂T /∂n = −fq /κ.

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Discretization

The Finite Volume Method

Turbulence

Turbulence

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Turbulence and its modeling

Introduction, dimensionless form In CFD, a scaled dimensionless form of the Navier–Stokes equation is often used. We introduce new variables xi∗ =

xi , L

ui∗ =

ui , U

p∗ =

p , U 2ρ

∂ 1 ∂ = , ∂xi L ∂xi∗

∂ U ∂ = , ∂t L ∂t ∗

in which the incompressible Navier–Stokes equations read ∇∗ · u∗ = 0, ∂u∗ ν + (u∗ · ∇∗ )u∗ = −∇∗ p ∗ + ∇2 u∗ . ∂t ∗ UL |{z} Re−1

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Turbulence and its modeling

Reynolds number and turbulence The Reynolds number is defined as Re =

UL Inertial forces = , ν Viscous forces

where U, L, ν are the characteristic velocity, characteristic length, and the kinematic viscosity, respectively. Fluids with the same Reynolds number behave the same way.

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Turbulence and its modeling

Reynolds number and turbulence The Reynolds number is defined as Re =

UL Inertial forces = , ν Viscous forces

where U, L, ν are the characteristic velocity, characteristic length, and the kinematic viscosity, respectively. Fluids with the same Reynolds number behave the same way. When the Reynolds number becomes larger than a critical value, the formerly laminar flow changes into turbulent flow, for example at Re ≈ 2300 for pipe flows.

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Turbulence and its modeling

Characterization of turbulence

Turbulent flows are I

time dependent

I

three dimensional

I

irregular

I

vortical (ω = ∇ × u)

Image from http://en.wikipedia.org

Describing the turbulence can be done in many ways, and the choice of the method depends on the application at hand and on computational resources. Martin Kronbichler (TDB)

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Turbulence and its modeling

Direct Numerical Simulation (DNS)

I

All scales in space and time are resolved.

I

No modeling of the turbulence.

I

Limited to small Reynolds numbers, because extremely fine grids and time steps are required, O(Re3 ) spatial and temporal degrees of freedom.

Applications of DNS: I

Turbulence research.

I

Reference results to verify other turbulence models.

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FVM for CFD

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Turbulence and its modeling

Large Eddie Simulation (LES)

Only the large scales are resolved, and small scales are modeled. Active research. LES quite popular in certain industries, but still very costly.

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FVM for CFD

February 11, 2010

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Turbulence and its modeling

Reynolds-Averaged Navier–Stokes (RANS) I Concept introduced by Reynolds in 1895. The most commonly used approach in industry. Only time averages are considered, nonlinear effects of the fluctuations are modeled. Use the decomposition ¯ (x, t) + u0 (x, t) , u(x, t) = u | {z } | {z } time average

fluctuation

where the time average is ¯ (x, t) = u

1 δ

Z

t+δ

u(x, τ ) d τ, t

(spatial averaging or ensemble averaging can otherwise be used).

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FVM for CFD

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Turbulence and its modeling

Reynolds-Averaged Navier–Stokes (RANS) II Continuity equation for averaged quantities: ∇ · u = 0. Momentum equations: 1 ∂u + (u · ∇)u + (u0 · ∇)u0 = − ∇p + ν∇2 u + f, ∂t ρ or, equivalently,   1 ∂u 0 0 + (u · ∇)u = ∇ · − pI + ν∇u − u ⊗ u + f, ∂t ρ

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FVM for CFD

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Turbulence and its modeling

Reynolds-Averaged Navier–Stokes (RANS) III

  ∂u 1 + (u · ∇)u = ∇ · − pI + ν∇u − u0 ⊗ u0 + f, ∂t ρ ∇ · u = 0. The Reynolds stress term −u0 ⊗ u0 contains fluctuations, and its effect needs to be modeled. Examples are Spalart-Allmaras, K − ε, K − ω, and SST.

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FVM for CFD

February 11, 2010

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