Governing equations Flashcards
Newton vs non-Newtonian
Newtonian - shear stress is directly proportional to rate of shear
non-Newtoninan - viscosity is a function of shear rate
What is CFD?
Combo of applied maths, computer science & fluid mechanics
CFD pros & cons
A: relatively cheap, detailed & consistent results, for complex problems
D: assumptions, needs validation, approximation
Approaches to analysing a fluid problem
Analytically (pure theory) - simple problem
Experimentally - validates analytical & simulation
Simulation (CFD) - complex problem
Physical principles that govern any fluid flow & what isn’t considered
- 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
CFD analysis steps
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)
Models of flow
Eulerian (conservation) - CV or element in fixed space
Lagrangian (non-conservation) - CV or element moves such that fluid particles are same (velocity - local velocity)
Body/volume force
act on element at a distance e.g. gravitational, electric, magnetic
Surface force
act on surface of element e.g. pressure (particle collision - thermodynamic pressure) & viscous (shear/normal, friction)
Momentum equation
L4 p22
Transient/unsteady
Convection (transport of fluid in space)
Source/sink (pressure gradient)
Diffusion (transport of fluid due to viscosity)
Body force
Continuity equation in conservation & non-conservation form in different notations
See workbook, L3 p18
Different derivatives & meanings
- 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)
Stokes relationship
expresses viscosity in relation to strain
relationship between viscous stresses & velocity gradients
Energy equation
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
Laminar vs turbulent
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)
Turbulence flow characteristics
- 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)
Large vs small eddies
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
Methods of numerical solution for turbulent flows
- 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
Turbulent boundary layer
1) Free stream
2) Viscous region (outer layer, fully-turbulent region/log-layer, buffer layer, viscous sublayer)
3) Wall
Universal velocity profile (law of the wall)
No matter fluid type/boundary layer, velocity profile is universal on graph (close to wall so not as affected by geometry)
How to obtain time-averaged momentum equations
1) Reynolds decomposition - split instantaneous velocity & pressure into mean + fluctuation components
2) Take time averaging
Where do Reynolds/turbulence stresses come from & what are they?
From non-linear convection terms
They’re normal & shear stresses due to turbulence
Why CFD needs validation
- doesn’t solve from 1st principles but from turbulence
- many assumptions
- set up of mesh not physically accurate
- uses discretised equations
What’s the closure problem?
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