CFD Flashcards
what the coupling between two phases can be
Dilute systems:
1 way coupling if the continuous phase affect the dispersed
2 way if also the dispersed phase affect the continuous one
Dense systems:
4 way if the dispersed phase interact with itself (when alpha_p>1e-3
how to evaluate phase coupling
with the particle response time
tau_v = rho_dd^2/(18mu_c)
tau_f = L_c/U_c
stokes = tau_v/tau_f
St«1 : 1 way coupling
St»1 : 2 way coupling
What is the geometry of a finite volume?
Each cell store the value of the quantity at cell centroid P
Faces and face center are called f
vector between centroid is d
d intersection with face f is f_i
Pf is the vector from P to face center f
S_f is the normal from face center f and his magnitude is the area of the face
if d ==S_f the mesh is orthogonal
transport equation
unsteady + convective = diffusion + source
look on notes for form
I use gauss on the convective and diffusion
I average convective, diffusion and source terms
How does the code solve the trasport equations and how does it calculate the values at the interface?
To get the value at faces I can use linear approximation, upwind of first or second order, quick or other methods.
If mesh is not ortogonal the solver will deal with it somehow (each solver is different on that)
At this point I discretize in space and time and build a matrix for each cell that will solve for the desired values.
The more ortogonal the mesh, the more diagonally dominated the matrix will be, the faster the solver would be
mesh types
in 2d: triangular, quadrilateral or polygonal
in 3d: thetraedral, pyramidal, prismatic, hexahedral, polyhedral
Structured vs unstructured mesh
Structured mesh are:
- vertical and horizontal lines crosses one time
- for simple geometry
- faster to build
- slower to get same accuracy as unstructured for same number of cell
- diagonal dominate matrix
Unstructred mesh are:
- for complicated geometries
- can change dimension in different part of the domain
- slower to build
- better accuracy
- sparse matrix
hexahedral vs thetahedral
Hexahedral (cubes):
- fewer elements
- faster
- more accurate
- naturally isotropic
- better for boundary layers
Thetahedral:
- better for complex geometries
- better mesh quality
- can connect smoothly different cell size
CFD vs FEA
CFD needs a boundary layer and a cell refinement towards the obstacles
skewness
Sk = max(sigma_max-sigma_e/(180-sigma_e), sigma_e-sigma_min/sigma_e)
sigma_e = 60° for triangles; 90° for quadrilateral
for triangles and thetahedrons:
Sk = optimal.cell.size-cell.size/optimal.cell.size
where the optimal is the inscribed in the same circunference
Sk = 0: GOOD
Sk = 1: BAD
Aspect ratio
for triangles and thetahedrons:
AR = f*(R/r)
f = 1/2(triangle) ; 1/3(thetahedron)
R = circumscribed circumference
r = inscirbed circunference
for squares and hexahedrons:
AR = max(e1, e2, …, en)/min(e1, e2, …, en)
e1 = (a+c)/2 ; e2 = (b+d)/2, …
n = 2 (2D) ; 3 (3D)
Ortogonal quality
for each phase is calculated:
-PfSf/(abs(Pf)abs(Sf))
-Sfd/(abs(Sf)abs(d))
OQ = min(previous value for each face)
OQ = 1: GOOD
OQ = 0 : BAD
Size ratio
SR = (dx_max/dx_min - 1) %
where dx are the length of 2 adiacent cells in the same direction.
SR should be lower than 20%
kolmogorv theory
It’s based on the creration of energy at big eddy scale and dissipation at small eddy scale.
Hp of local isotropy: for high Re, the small scale is statistically isotropic
1 Hp of similarity: for high Re, small scale is described by nu and eps.
2 Hp of similarity: for high Re, intermediate scale is described by eps.
Energy at wavelenght k is E(k) = Ceps^(2/3)k^(-5/3)
Reynolds stress
It emerges in momentum equation when substituing the velocity with the average plus fluctutation.
tau_ij = - avg(u_iu_jrho)
It introduce 6 unkonwn that needs to be modeled.
Baussinesq hp says that they are proportional to mu_t velocity gradient + k, where mu_t = Costrhok^2/eps is the turbulent viscosity and k = 0.5(avg(x’^2)+avg(y’ì2)+avg(z’^2)) is the turbulent kinetic energy per unit mass
possible models for turbulence transport equation
k-eps standard: simple, stable and robust with wall function.
k-eps RNG: better for swirl
k-eps realizable: more accurate adn easier to converge than RNG
ALL k-eps are bad near wall and for separating flows
omega = eps/k
k-omega standard. good for walls but bad away from walls. good for separating flows, low Re, compressibility. very used in aerospace and turbomachines
k-omega SST: merge k-omega and k-eps to get the best of both
Introduction to wall treatment
Near the wall properties like velocity or temeperature change very rapidly. Friciton with the wall needs to be properly calculated. The wall sheer strees is required (tau_w)
We can model the boundary layer (wall function treatment) or resolve the boundary layer (near wall treatment)
effective wall viscosity and first step for obtaining a wall function
when refining the mesh is too complicated a function is used to describe the values at the wall. the real wall sheer stress is:
tau_w = -rhonu(dU/dy) for y=0
but we can only calculate the value in the centroid P for U:
tau_w,cfd = -rhonuU_P/y_P.
We correct nu as nu_w, effective wall viscosity
Using U+ = U_P/u_tau = fn(y+)
y+ = rhou_tauy/mu
u_tau = sqrt(tau_w/rho)
I get nu_w = u_tau*y_P/fn(y+)
But what is fn(y+)?
y plus value, U+ = ?, when y+ function is good?
U+, y+ is the same for all flow because they are adimensional
y+ = rho* u_tau* y/mu
U+ = y+ for y+<11.25
U+ = constln(consty+) for y+>11.25
not good in
y+ > 200 becuase of bulk quantities
y+ < 5 because of degradation of turbulent equation near the wall
y+ = [5,30] because of transition from linear to logaritmic
solving the boundary layer with the near wall treatment
Can be implemented for k-eps (enhanced wf) and it’s default for k-omega (automatic wf).
It uses a function for calculating f(y+).
It’s best to avoid buffer region.
It’s best to have y+<3 and 10 cells
for high y+ it just revert to a log law wall function
What is Population Balance Model
A PBM is described by a transport equation for the particle size distribution, with an uinsteady term, a diffusion term and a source term.
The source term include: coalescence and break up, plus other term like mass transfer or other that are often not considered.
For coalescence we need to model the frequency and efficency with the film drainage model, for breakup we want the frequency and the daughter size distribution.
Different dispersed size models
Adiabatic fixed mono-dispersed
Adiabatic fixed poly-dispersed
Adiabatic poly-dispersed with coalescence and breakup
homogeneous vs inhomogeneous PBM
Bins are created on the bubble size.
In homogeneous balance model a continuity equation is solved for each bin of the secondary phase but the velocity field is the same for all.
In inhomogeneous balance model both continuity and momentum balance equation are done, but for different size of bins to not increase too much the compuitational load. Slower but more precise.
Describe the DPM
The discrete phase model tracks the particles with a lagrangian reference frame so it solves continuity, momentum and energy for the continuous phase wihle only momentum and energy for the dispersed phase.
The key HP:
- particle does not displace volume
- particle is a sphere
-1 or 2 way coupling (alfa_p<0.1). It’s important to check the stokes number
-particle parcel concept
How can particle size be calculated in DPM
In discrete phase model the particel size can be calculated from:
- statistics (such as log-normal distribution or roslin-rammler distribution)
- sauter mean
- experimental data.
Introduction to VOF and what is the interface
The VOF method wants to track the interface between two fluids. Phisically the interface is a region of space, nanometers order, that where the properties changes continuously but in CFD this change is abrupt. The acting forces at the interface are the surface tension andthe latent heat of vaporization.
VOF: one fluid formulation
The two fluids are actually computed as a single one and the change in properties is calculated with a Indicator function, chanign from 0 to 1. The interface can be found as the gradient of this function. The continuity, momentum and energy equation can now be written with a force term due to the surface tension, active only at the interface.
VOF: volume fraction formulation
Actually in VOF the indicator function is the volume fraction, ranging from 0 to 1 at the interface. So a transport equation for the volume fraction needs to be specifically solved. This allows a easier and more accurate implementation but has the problem of adding a new equation that needs to be solved.
VOF: geometrical and algebraic advection
Solving the transport for the volume fraction is problematic since 1st order scheme are diffusive while higher order scheme introduce oscillations.
A geometrical scheme that tracks the interfaces can be implemented that allows mass conservation but is computationally expensive
A algebraic scheme can be implemented, solving one space varibale at a time and using a discretization in time and space. It’s lower computationally but it doesn’t conserve mass
VOF: surface tension force
Calculating the proper surface tension is tricky because of numerical divergence which can hppen in some points of the domain. Smoothing the interface or using the Height function can help.
VOF: phase change
It’s complicated to calculate the heat exchange at the interface. The most popular approaches are:
-thermal equilibrium at the interface
-thermal equilibrium at the interface according to Lee’s model
-departure from thermal equilibrium
The typical system studied are:
- evaporation so surface is moving
- heat transfer
- bubble grow in superheated liquid
The difficulty stands in modeling phenomena that starts at a microscale.
