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CFD > Governing equations > Flashcards

Governing equations Flashcards

1
Q

Newton vs non-Newtonian

A

Newtonian - shear stress is directly proportional to rate of shear
non-Newtoninan - viscosity is a function of shear rate

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2
Q

What is CFD?

A

Combo of applied maths, computer science & fluid mechanics

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3
Q

CFD pros & cons

A

A: relatively cheap, detailed & consistent results, for complex problems
D: assumptions, needs validation, approximation

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4
Q

Approaches to analysing a fluid problem

A

Analytically (pure theory) - simple problem
Experimentally - validates analytical & simulation
Simulation (CFD) - complex problem

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5
Q

Physical principles that govern any fluid flow & what isn’t considered

A
  • Conservation of mass (continuity)
  • Conservation of momentum (F=ma)
  • Conservation of energy (1st law of thermodynamics, Bernoulli)
  • Governed by Navier-Stokes equations (mathematical model)
  • Not considering mass diffusion due to concentration gradients/chemically reacting flows
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6
Q

CFD analysis steps

A

1) Understand physics
2) Mathematical model (equations, BCs, turbulence modelling, near wall models)
3) Numerical model - geometry, mesh generation, FVM, fluid properties
4) Solution - set numerical parameters, solve discretised equations (matrix inversion/iteratively)
5) Post-processing - verify, validate (experiment)

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7
Q

Models of flow

A

Eulerian (conservation) - CV or element in fixed space
Lagrangian (non-conservation) - CV or element moves such that fluid particles are same (velocity - local velocity)

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8
Q

Body/volume force

A

act on element at a distance e.g. gravitational, electric, magnetic

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9
Q

Surface force

A

act on surface of element e.g. pressure (particle collision - thermodynamic pressure) & viscous (shear/normal, friction)

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10
Q

Momentum equation

A

L4 p22
Transient/unsteady
Convection (transport of fluid in space)
Source/sink (pressure gradient)
Diffusion (transport of fluid due to viscosity)
Body force

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11
Q

Continuity equation in conservation & non-conservation form in different notations

A

See workbook, L3 p18

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12
Q

Different derivatives & meanings

A
  • substantial/total (net temperature difference due to change in space & time)
  • local (temperature change due to change in time at a fixed location)
  • convective (temperature change due to movement in space)
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13
Q

Stokes relationship

A

expresses viscosity in relation to strain
relationship between viscous stresses & velocity gradients

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14
Q

Energy equation

A

Flow model: fluid element moving with fluid (Langrangian - non-conservation)
rate of change of energy = net heat flux + work done on element
U = Q + W

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15
Q

Laminar vs turbulent

A

Laminar - viscous dominant over inertial, parabolic curve, NS can be solved numerically
Turbulent - NS only solved for low-Re simple geometry flows
Influenced by Re, surface roughness, geometry, pressure gradients, ambient disturbances (vibrations)

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16
Q

Turbulence flow characteristics

A
  • Irregular fluctuation (but not statistically random, a few % around mean value)
  • Enhanced diffusivity (mixing, larger rate of transport of mass, heat, momentum)
  • rotational vortex tubes (of different sizes & scales superimposed on each other)
  • energy dissipation (viscous shear stress converts kinetic energy of turbulence into internal energy of fluid)
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17
Q

Large vs small eddies

A

Large - length comparable to flow field e.g. pipe radius, boundary layer thickness
Small - several orders of magnitude smaller than largest eddy but much larger than molecular mean free path, most energy dissipation occurs in smallest eddies

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18
Q

Methods of numerical solution for turbulent flows

A
  • Direct Numerical Solution (DNS) - solves exact N-S eq.s, mesh must be small enough for smallest dissipative scales - for research & low-Re flows (less eddies otherwise need too many cells)
  • Large eddy simulations (LES) - solves filtered N-S eq.s, models small scale turbulence (mesh <=resolved turbulence scales) & resolves large turbulence scales by low-pass filtering to remove small-scale info - next generation engineering tools
  • Reynolds Averaged Navier Stokes equations (RANS) - empirical turbulence models where instantaneous velocity decomposed into time-averaged (mesh resolves mean flow) & fluctuating component - widely used, CFD solves RANS not NS
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19
Q

Turbulent boundary layer

A

1) Free stream
2) Viscous region (outer layer, fully-turbulent region/log-layer, buffer layer, viscous sublayer)
3) Wall

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20
Q

Universal velocity profile (law of the wall)

A

No matter fluid type/boundary layer, velocity profile is universal on graph (close to wall so not as affected by geometry)

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21
Q

How to obtain time-averaged momentum equations

A

1) Reynolds decomposition - split instantaneous velocity & pressure into mean + fluctuation components
2) Take time averaging

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22
Q

Where do Reynolds/turbulence stresses come from & what are they?

A

From non-linear convection terms
They’re normal & shear stresses due to turbulence

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23
Q

Why CFD needs validation

A
  • doesn’t solve from 1st principles but from turbulence
  • many assumptions
  • set up of mesh not physically accurate
  • uses discretised equations
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24
Q

What’s the closure problem?

A

From RANS equations 4 eq.s, 10 unknowns including 6 turbulence stresses (normal & shear stresses)
Transport equations can be derived for Reynolds stresses but more unknowns will appear
Solution: develop empirical turbulent models to approximate turbulence stresses

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25
Q

What is the eddy viscosity concept?

A

Analogy between viscous & turbulent stresses - both mixing & friction
Viscosity is fluid property
Turbulent viscosity is not, but depends on flow (much greater than viscosity in turbulent flow)
Most turbulence models are based on this concept incl. k-epsilon model & on Prandtl hypothesis

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26
Q

Prandtl hypothesis

A

Based on dimensional analysis
Turbulent/eddy viscosity can be approximated as characteristic turbulence velocity*turbulence length scale
Different ways of defining turbulence velocity & length scale lead to different turbulence models

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27
Q

How are the model constants obtained in the k-epsilon turbulence model?

A

Tuned using simple flows & experimental data e.g. near wall flow in local equilibrium/grid generated turbulence

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28
Q

Story of the k-epsilon turbulence model

A

1) RANS
2) Eddy viscosity concept (Boussinesq hypothesis)
3) Turbulent/eddy visocisty
4) transport equations for k & e
5) model constants
time variation & convection = production - dissipation + diffusion (molecular & turbulent)

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29
Q

Classification of turbulence models

A

1) Eddy viscosity models - linear/non-linear (add function of strain rate & omega & is bridge between linear & RSMs)
2) Reynolds stress turbulence models (RSMs) - linear/quadratic pressure-strain RSM models (directly solves turbulence stress)

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30
Q

Qualities of a good model

A

accuracy, simplicity, robustness (numerically), breadth of applications, economic to run, aligns to physics of flow

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31
Q

Why is the physics of flow important when choosing a CFD model?

A

most models are based on linear eddy viscosity & are developed & calibrated for simple wall shears so some physics of round jet might not be captured by these models - there are additional corrections to models to account for these

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32
Q

Standard k-epsilon vs K-omega SST vs RSM

A

Epsilon - very robust & economic, most widely used but not most accurate (free stream turbulence not dominating flow behaviour)
Omega - becoming popular as deals with complex flows but more expensive & less robust
RSM - for specific complex flows but very expensive & not robust (difficult to converge) (flow dominated by swirl)

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33
Q

0 equation/algebraic

A

A: easy to implement, fast calculation, good predictions for simple flows where experimental correlations for the mixing length exist, used as part of some higher-order models
D: incapable of describing flows where turbulent length scale varies (separation/recirculation), only calculates mean flow properties & turbulent shear stress, can’t switch from 1 type of region to another, history effects of turbulence not considered
Usage: derives analytical expressions for turbulent flows, for simple external aero flows, ignored in commercial CFD programs

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34
Q

1-equation model

A

A: attached wall-bounded flows, flows with mild separation & recirculation, for unstructured codes in aero, in aeronautics for computing flow around plane wings
D: massively separated flows, free shear flows, decaying turbulence, complex internal flows. Can’t rapidly accommodate changes in length scale
Models: Spalart-Allmaras, k-model, Baldwin-Barth

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35
Q

Standard k-epsilon model

A

A: simple, robust, widely applicable, easy convergence
D: poor for swirling/strong separation/jets/unconfined flows/non-circular ducts. Insensitive to adverse pressure gradient, predicting delayed separation
Usage: only valid for fully turbulent flows, requires wall function implementation, for internal flows, overpredicts production of turbulence in highly strained flows

36
Q

RNG k-epsilon

A

Derived with statistics (renormalisation group method) to instantaneous N-S eq.s
Improved predictions for curvature, strain rate, transitional & separated flows, wall heat & mass transfer, vortex shedding but NOT spreading of a round jet

37
Q

Realizable k-epsilon

A

satisfies certain maths constraints on Reynolds stresses, consistent with physics of turbulent flows
A: avoids -ve normal stresses & violation of Schwarz inequality for shear stresses, improved e equation, variable c_viscosity
Improved performance for jet spreading, boundary layers, separation, pressure gradient, rotation, recirculation, strong streamline curvature

38
Q

k-omega

A

A: better for transitional flows & flows with adverse pressure gradients & separation, numerically very stable, low-Re version more economical
D: not very consistent, sensitive to free-stream boundary conditions

39
Q

SST k-omega

A

Combines k-omega (inner boundary layer) & k-epsilon (outer regions), limits shear stress in adverse pressure gradient regions
Avoids freestream sensitivity
Is a wall resolved model so does not need a wall function
Widely used for aerodynamic flows

40
Q

non-linear turbulence model

A
  • accounts for effects such as
    streamwise curvature, impinging jet, anisotropy
    not widely used in practice
41
Q

RSM

A
  • avoids isotropic eddy viscosity assumption
  • accurately predicts complex flows by accounting for streamline curvature, swirl, rotation & high strain rates
  • includes turbulent pressure-strain interactions & rotation
    6 PDEs for each of the 6 independent Reynolds stresses
  • more expensive & difficult to converge
42
Q

Near-wall modelling strategies

A

1) wall-function/standard/high-Re turbulence models - bridges wall & outer region (30<yp+<300) - standard wall function/non-equilibrium wall function
2) wall-resolved turbulence model - directly solves near-wall flow (yp+<1), considers viscous, high strain rate & wall effects - low-Re model/2-layer zonal model (high-Re in ouer & low-Re near wall)
3) Enhanced wall treatment - combines 2-layer zonal model with advanced wall function (1<yp+<30)

43
Q

Standard wall function

A

A: economical, robust, reasonably accurate for many applications, widely used
D: adverse pressure gradients, flow separation, strong body force, strong 3D effects near wall region, pervasive low Re e.g. flow through small gap, highly viscous, low velocity flow

44
Q

Non-equilibrium wall function

A
  • 2 layer-based concept computes k & e
    A: log-law of wall made to sensitise pressure-gradient or improved predictions of separation, reattachment, impingement
    D: poor for low-Re effects, massive transpiration (blowing, suction), sever pressure gradient, strong body forces, highly 3D flows
45
Q

low-Re model

A

not for low-Re flows but for wall-resolved modelling (local Re is low)
A: damping functions to consider viscous effect, high strain rate, wall
D: fine mesh & expensive

46
Q

2 layer zonal model

A

Inner: 1-equation low-Re model
Outer layer: high-Re model of choice
- Does not rely on wall functions
- Zones defined by wall-distance-based turbulent Reynolds number
- Flow within boundary layer calculated explicitly
- k-equation solved in viscosity-affected region
- epsilon computed using a correlation for turbulent length scale
- zoning may be dynamic & solution adaptive
D: fine mesh & expensive

47
Q

Enhanced wall treatment

A

epsilon-equation
only if mesh too coarse to resolve viscous sublayer
- in fully turbulent region (Re>200) - high-Re mode
- in viscosity affected near-wall region (Re<200) - 1-eq model of Wolfstein
1<yp+<30 but if high accuracy required e.g. heat transfer use y+<1 or y>30

48
Q

Numerical solution process

A

Numerical solutions for RANS which are PDEs
1) meshing - divide domain into a grid with nodes
2) discretisation - approximate PDEs using algebraic equations for nodal values of velocity, pressure etc.
3) Solution - solve linearised algebraic equations iteratively

49
Q

Finite difference method

A

Taylor series to discretise PDEs
A: can have higher orders of accuracy schemes, easy to apply for simple flow geometry using a structured mesh
D: not straightforward to apply to complex geometry, conservation not guaranteed due to truncation errors
Used in DNS CFD but not commericial CFD

50
Q

Finite volume method

A

1) flow domain divided into CVs
2) PDEs integrated over CVs
3) PDEs discretised
4) Obtain set of linear algebraic equations
5) Apply to each CV
6) Solve algebraic equations
A: for complex geometries with structured/unstructured mesh, ensures conservation
D: difficult to apply higher order schemes
Used in commercial research CFD

51
Q

FVM - how to evaluate source terms

A

evaluate at cell centre & multiply by volume of cell

52
Q

FVM - how to evaluate diffusion fluxes

A

Evaluate gradient at cell faces using central difference based on linear
2nd order & numerically stable

53
Q

FVM - how to evaluate convection fluxes

A

mass fluxes through east & west faces
Interpolate nodal values to get values of variable at the cell faces
Scheme to interpolate affects stability & accuracy

54
Q

Central difference scheme

A

linear interpolation between phi P & phi E to approximate phi e
phi P & phi E to approximate phi w
L12 & 13 p171
For convection schemes 2nd order but produces oscillatory solutions

55
Q

First-order upwind scheme

A

A: always bounded & stable
D: 1st order accuracy so fine grid needed for sufficient accuracy

56
Q

Second-order upwind scheme

A

2nd order accurate
not bounded - produces undershoots & overshoots in regions of steep gradients

57
Q

QUICK

A

quadratic upwind interpolation for convection kinetics
parabola of 3 points
3rd order accurate
not bounded

58
Q

Properties of interpolation schemes

A
  • accuracy - truncation errors
  • conservativeness - global conservation of fluid property must be ensured (particularly energy & mass)
  • boundedness - values predicted should be within realistic/physical bounds
  • transportiveness - scheme should reflect process e.g. convection follows flow direction so upstream influence dominates, diffusion works in all directions so upstream & downstream influence diffusion equally

Mesh refinement & higher order scheme minimises false diffusion & improves accuracy
Error in upwind scheme is equivalent to increasing diffusivity - stable but inaccurate
- accuracy requires numerical diffusion &laquo_space;physical diffusion

59
Q

Difference between space & time coordinates

A

forcing affects all spatial directions - elliptical
forcing on affects future - parabolic (solutions march in time)

60
Q

Explicit Euler method

A

approximate f at initial time
A: cheap
D: 1st order accurate, instable

61
Q

Courant/CFL number

A
  • requirement for achieving a stability solution
  • ratio of time step to time required for flow info to be convected across cell
  • time step must be sufficiently small so a fluid element cannot travel more than 1 grid cell length in each time step
  • Smaller time step required when mesh is refined
62
Q

Fully implicit method

A
  • approximate integral using value of f at final time
    A: unconditionally stable (may still be unstable if time step too big)
    D: first order accurate, requires iterative procedure, must solve large coupled system of equations at each time step, more expensive
63
Q

Crank-Nicolson scheme

A
  • approximate integral using weighted average of initial & final values of f using trapezium rule
    A: 2nd order accurate, larger time steps can be taken whilst retaining acceptable temporal accuracy
    D: implicit scheme so requires iterative procedure
  • Von-Neumann analysis shows it’s unconditionally stable but in practice becomes unstable for very large time steps
64
Q

Solution methods

A

1) Pressure-based solvers (U, V, W, P) - incompressible flows - segregated/coupled
2) Density-based solvers (U, V, W, density) - compressible flows - coupled algorithm

65
Q

What is the pressure-velocity coupling problem?

A

don’t explicitly have an equation for pressure

66
Q

Pressure-based solver - segregated - types

A

SIMPLE (semi-implicit method for pressure-linked equations)
SIMPLER (revised - default but more computational effort)
SIMPLEC (consistent - more computational effort)
PISO (pressure implicit with splitting of operators - initially developed for unsteady flows, involves 2 correction stages)

67
Q

Solution procedure for pressure-based segregated solver

A

1) guess a pressure field
2) solve momentum equations for velocity fields (u, v, w) sequentially using this P
3) solve pressure-correction (continuity) equation (mass conservation)
4) calculate correction for velocities
5) update/correct pressure, velocities & mass flux
6) sequentially solve energy, species, turbulence & other scalar equations
7) repeat until convergence

  • long solution time (due to pressure correction algorithm) but moderate memory required
68
Q

How does a density-based solver work?

A

1) equation of state P = (density, T) links P and density
2) density treated as primary variable & solved from continuity equation
3) P calculated from equation of state
4) solve continuity, momentum, energy & species equations simultaneously
5) solve turbulence & other scalar equations sequentially
6) repeat until convergence

  • density appears in continuity equation
  • energy equation also needs to be solved for T
  • compressible flow when M>0.3
  • efficient for compressible flow, especially shock waves
  • large memory & computing power but solution time less dependent on mesh density
69
Q

Solution procedure for pressure-based coupled solver

A

1) Simultaneously solve system of momentum & pressure-based continuity equations
2) Update mass flux
3) Sequentially solve energy, species, turbulence & other scalar equations
4) Repeat till converged

  • more memory but potentially converges faster
70
Q

Solution of linear algebraic equations

A

RANS equations are non-linear & coupled
- Direct method - solve all equations at all nodes simultaneously (huge computer resources) - Cramer’s rule matrix inversion, Gaussian elimination (N^3 operations for N equations, store N^2 coefficients)
- Iterative method - solve equations at 1 node, then march through all nodes in solution domain (huge no. of iterations) - Jacobi, Gauss-Seidel (simultaneous storage of only non-zero equations coefficients)

71
Q

Gauss-Seidel/Jacobi

A

Gauss-Seidel - latest values of neighbour nodes
Jacobi - previous iteration
- repeat iteration until 2 successive iterations are smaller than convergence criterion/small residuals in equations
- slow convergence with large mesh (global errors reduce slower (low residual reduction) when mesh is fine (large no. of nodes)) > multigrid scheme accelerates convergence
- Gauss-Seidel rapidly removes local (high-frequency errors)

72
Q

Multigrid scheme

A

sequence of successively coarser meshes - iterate between solutions on coarse/fine meshes
- global errors reduce faster (less cells in coarse mesh) & less computing resources required, does NOT affect accuracy

73
Q

Under-relaxation factor

A

moderate changes in phi to avoid unstable iterations (slow convergence/divergence) due to non-linearity of equations. Use alpha% of phi, otherwise solution may not converge
- small value of alpha ensures convergence but costs more iterations
- no general rules for optimum value of alpha
- different variables require different values of alpha
- Relaxation 1st thing to look at if iteration diverged

74
Q

Geometry generation

A
  • top-down (cylinders, bricks, spheres)
  • bottom-up (create faces & volumes)
  • import geometry from CAD/graphics
75
Q

mesh affects…, mesh is affected by ….

A
  • rate of convergence
  • solution accuracy
  • CPU time required
  • topology
  • density
  • quality
  • boundary layer mesh
  • refinement
76
Q

mesh topology

A
  • triangle/tetrahedron
  • quadrilateral/hexahedron/prism
  • structured
  • unstructured (memory, CPU overhead, more diffusive losing accuracy)
77
Q

mesh quality

A
  • skewness - angles between edges/faces
  • aspect ratio - longest edge length : shortest edge length (<=5:1)
  • smoothness (<1.2)
  • resolution
  • flow alignment (reduces truncation error)
78
Q

mesh for boundary layer

A
  • laminar - 5 cells
  • turbulent wall-function - 30<yp+<300
  • viscous sub-layer resolved - yp+=1 & viscous sublayer contains 5 cells, boundary layer has 10-20 cells
79
Q

mesh resolution

A

to capture geometry, use finer mesh in regions of shear, swirling flow
- min 5 cells across shear layers
- concentrate mesh around vortex cores
- ensure mesh resolution is sufficient upstream of region of interest
- at least 3 nodes between 2 walls

80
Q

Why refine mesh?

A
  • improve resolution of flow features
  • should not invalidate turbulence model/alter solution
  • mesh dependence study
81
Q

non-conformal mesh

A

neighbouring cells are not 1 to 1
- for large/moving mesh problems
- interface must be sufficient distanced from region of interest

82
Q

What is a residual?

A

error in solving conservation equations

  • absolute residual
  • scaled residual (relative error) - relative to local value of phi
  • normalised error - divide current residual by residual at step N to calculate how it’s reduced relative to previous steps
  • global scaled residual - compare with all cells in domain
83
Q

Residuals monitor - what do you look at?

A
  • convergence for scaled/normalised residual
  • monitor changes in value of phi with iterations at critical points in flow
84
Q

Flow parameter monitoring

A
  • choose parameter at critical point/point of interest in flow field e.g. coefficient of friction
85
Q

Convergence problems

A
  • check problem setup (BCs, direction of gravity, physical model)
  • poor mesh quality, y+ values
  • poor initialisation values, especially T/density
  • strong coupling of flow & scalar variables e.g. combustion
  • solve complex multi-physics problems in stages
  • reduce relaxation factors/time step
  • use simpler model for initial guess to complex model
86
Q

Verification

A
  • solve equations right (accuracy of CFD model & implementation/setup)
  • machine round-off error (double precision)
  • iterative convergence error (truncation error, 1st or 2nd order, convergence criteria)
  • discretisation error (mesh dependence study)
87
Q

Validation

A
  • solve right equations
  • input uncertainty - sensitivity analysis (different BCs)
  • model uncertainty - turbulence model, compare with experimental results
  • qualitative validation - inspect basic flow physics

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