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TurbAna.py
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TurbAna.py
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"""
Turbulence Analyzer (TurbAna) Python toolkit Ver 1.0
The script was originally written for analyzing the turbulence anisotropy of
compressor tip leakage flows [1] and the eddy viscosity field of transonic
bump flows [2]:
[1] He, X., Zhao, F., & Vahdati, M. (2022). Detached Eddy Simulation: Recent
Development and Application to Compressor Tip Leakage Flow. ASME Journal
of Turbomachinery, 144(1), 011009.
[2] He, X., Tan, J., Rigas, G., and Vahdati, M. (2022). On the Explainability
of Machine-Learning-Assisted Turbulence Modeling for Transonic Flows.
International Journal of Heat and Fluid Flow, 97, 109038.
An explict reference to the works above is highly appreciated if this script is
useful for your research.
Xiao He (xiao.he2014@imperial.ac.uk)
Last update: 20-August-2022
"""
# -------------------------------------------------------------------------
# import libraries
import numpy as np
import matplotlib.pyplot as plt
from scipy import linalg
# -------------------------------------------------------------------------
# main classes
class MeanFlowField:
'''
Purpose: class of mean flow field
Parameters
----------
MeanFlow: 2D numpy array, float.
nrow = ngrid (number of grid)
ncol = nvar (e.g., p, T, u, v, w, mut)
'''
def __init__(self, MeanFlow=np.ones([1,6])):
try:
[ngrid, nvar] = MeanFlow.shape
self.ngrid = ngrid
self.nvar = nvar
if nvar == 4: # 2D data with p, T, u, v
print('Initialize MeanFlowField using 2D data...')
MeanFlow = np.concatenate((MeanFlow, np.zeros(shape=[ngrid, 2])), axis=1)
elif nvar == 5: # 2D data with p, T, u, v, mut
print('Initialize MeanFlowField using 2D data...')
MeanFlow = np.concatenate((MeanFlow, np.zeros(shape=[ngrid, 1])), axis=1)
elif nvar == 6: # 3D data with p, T, u, v, w, mut
print('Initialize MeanFlowField using 3D data...')
else:
raise ValueError
except:
raise ValueError('Wrong data type: please use 2D numpy array with ncol = 4 or 9 for MeanFlowField')
# initialize values
self.p = MeanFlow[:,0]
self.T = MeanFlow[:,1]
self.Velocity = MeanFlow[:,2:5]
self.EddyVisc = MeanFlow[:,5]
self.LamVisc = self.CalcLamVisc
self.Density = self.CalcDensity
def __str__(self):
return("Mean flow field of %.i grid points."%(self.ngrid))
@property
def CalcLamVisc(self):
'''
Purpose: calculate laminar viscosity using Sutherland's law
'''
mu_ref = 1.716 * 10**(-5)
Tc =273.15
S = 110.4
mu = mu_ref*(self.T/Tc)**1.5*(Tc+S)/(self.T+S)
return(mu)
@property
def CalcDensity(self):
'''
Purpose: calculate air density using the ideal gas equation
'''
R=287
rho = self.p/R/self.T
return(rho)
class MeanGradField:
'''
Purpose: class of mean gradient field
Parameters
----------
MeanGrad: 2D numpy array, float.
nrow = ngrid (number of grid)
ncol = nvar (e.g., dudx, dudy, dvdx, dvdy for 2D;
dudx, dudy, dudz, dvdx, dvdy, dvdz, dwdx, dwdy, dwdz for 3D)
'''
def __init__(self, MeanGrad=np.ones([1,9])):
try:
[ngrid, nvar] = MeanGrad.shape
self.ngrid = ngrid
self.nvar = nvar
if nvar == 4: # 2D data with dudx, dudy, dvdx, dvdy
print('Initialize MeanGradField using 2D data...')
MeanGrad = np.concatenate((MeanGrad[:,0:2],np.zeros(shape=[ngrid, 1]),
MeanGrad[:,2:4],np.zeros(shape=[ngrid, 4])), axis=1)
elif nvar == 9: # 3D data with
print('Initialize MeanGradField using 3D data...')
else:
raise ValueError
except:
raise ValueError('Wrong data type: please use 2D numpy array with ncol = 4 or 9 for MeanGradField')
# initialize values
self.VelGrad = MeanGrad
self.StrainRate = self.CalcStrainRate
self.Vorticity = self.CalcVorticity
self.StrainRateMag = self.CalcStrainRateMag
self.VorticityMag = self.CalcVorticityMag
def __str__(self):
return("Mean gradient field of %.i grid points."%(self.ngrid))
@property
def CalcStrainRate(self):
'''
Purpose: calculate strain rate tensor components (Sxx, Syy, Szz, Sxy, Sxz, Syz)
'''
Sxx = self.VelGrad[:,0]
Syy = self.VelGrad[:,4]
Szz = self.VelGrad[:,8]
Sxy = 0.5*(self.VelGrad[:,1]+self.VelGrad[:,3])
Sxz = 0.5*(self.VelGrad[:,2]+self.VelGrad[:,6])
Syz = 0.5*(self.VelGrad[:,5]+self.VelGrad[:,7])
S = (np.array([Sxx,Syy,Szz,Sxy,Sxz,Syz])).T
return(S)
@property
def CalcVorticity(self):
'''
Purpose: calculate vorticity tensor components (Oxy, Oxz, Oyz)
'''
Oxy = 0.5*(self.VelGrad[:,1]-self.VelGrad[:,3])
Oxz = 0.5*(self.VelGrad[:,2]-self.VelGrad[:,6])
Oyz = 0.5*(self.VelGrad[:,5]-self.VelGrad[:,7])
O = (np.array([Oxy,Oxz,Oyz])).T
return(O)
@property
def CalcStrainRateMag(self):
'''
Purpose: calculate strain rate magnitude Smag
'''
Smag = np.sqrt(2)*np.sqrt(self.StrainRate[:,0]**2+self.StrainRate[:,1]**2+
self.StrainRate[:,2]**2+2*self.StrainRate[:,3]**2+2*self.StrainRate[:,4]**2+
2*self.StrainRate[:,5]**2)
return(Smag)
@property
def CalcVorticityMag(self):
'''
Purpose: calculate vorticity rate magnitude Omag
'''
Omag = 2*np.sqrt(self.Vorticity[:,0]**2+self.Vorticity[:,1]**2+
self.Vorticity[:,2]**2)
return(Omag)
def CalcVelGradMag(self):
'''
Purpose: calculate the F-norm of the velocity gradient tensor
'''
ugradmag = np.sqrt(np.sum(self.VelGrad**2, axis=1))
return(ugradmag)
def CalcTensorBasis(self,nond_weight=False,cap=7,norm=False):
'''
Purpose: calculate the tensor invariants and tensor basis of Pope's "General Effective-Viscosity Hypothesis"
Parameters
----------
nond_weight: 2D numpy array, float.
nrow = ngrid (number of grid)
ncol = 1 (weight)
cap : min-max value of the non-dimensionalized S and O component, float.
'''
# Sij
S_trace = self.StrainRate[:,0]+self.StrainRate[:,1]+self.StrainRate[:,2]
Sij = np.zeros(shape=[self.ngrid,3,3])
Sij[:,0,0] = self.StrainRate[:,0]-S_trace/3
Sij[:,1,1] = self.StrainRate[:,1]-S_trace/3
Sij[:,2,2] = self.StrainRate[:,2]-S_trace/3
Sij[:,0,1] = self.StrainRate[:,3]
Sij[:,0,2] = self.StrainRate[:,4]
Sij[:,1,0] = Sij[:,0,1]
Sij[:,1,2] = self.StrainRate[:,5]
Sij[:,2,0] = Sij[:,0,2]
Sij[:,2,1] = Sij[:,1,2]
# Oij
Oij = np.zeros(shape=[self.ngrid,3,3])
Oij[:,0,1] = self.Vorticity[:,0]
Oij[:,0,2] = self.Vorticity[:,1]
Oij[:,1,0] = -Oij[:,0,1]
Oij[:,1,2] = self.Vorticity[:,2]
Oij[:,2,0] = -Oij[:,0,2]
Oij[:,2,1] = -Oij[:,1,2]
# non-dimensionalize Sij and Oij
if type(nond_weight) != bool:
for k in range(self.ngrid):
Sij[k,:,:] = Sij[k,:,:]*nond_weight[k,0]
Oij[k,:,:] = Oij[k,:,:]*nond_weight[k,0]
# min-max limiter of Sij and Oij
Sij[Sij > cap] = cap
Sij[Sij <-cap] =-cap
Oij[Oij > cap] = cap
Oij[Oij <-cap] =-cap
# re-enforce trace zero for Sij
for k in range(self.ngrid):
Sij[k,:,:] = Sij[k,:,:] - np.eye(3)*np.trace(Sij[k,:,:])/3
# lambdas
lambdas=np.zeros(shape=[self.ngrid,5])
for i in range(self.ngrid):
lambdas[i, 0] = np.trace(np.dot(Sij[i, :, :], Sij[i, :, :]))
lambdas[i, 1] = np.trace(np.dot(Oij[i, :, :], Oij[i, :, :]))
lambdas[i, 2] = np.trace(np.dot(Sij[i, :, :], np.dot(Sij[i, :, :], Sij[i, :, :])))
lambdas[i, 3] = np.trace(np.dot(Oij[i, :, :], np.dot(Oij[i, :, :], Sij[i, :, :])))
lambdas[i, 4] = np.trace(np.dot(np.dot(Oij[i, :, :], Oij[i, :, :]), np.dot(Sij[i, :, :], Sij[i, :, :])))
# normalize lambdas
if norm == 'min-max':
lambdas = (lambdas-np.min(lambdas,axis=0))/(np.max(lambdas,axis=0)-np.min(lambdas,axis=0))
elif norm == 'mean-std':
lambdas = (lambdas-np.mean(lambdas,axis=0))/np.std(lambdas,axis=0)
# Caps the max value of the invariants after first normalization pass
Icap = 2
for j in range(5):
nloop=0
while np.max(np.abs(lambdas[:,j]))>Icap:
lambdas[:,j]=np.clip(lambdas[:,j],-Icap,Icap)
lambdas[:,j]=(lambdas[:,j]-np.mean(lambdas[:,j]))/np.std(lambdas[:,j])
nloop+=1
if nloop>0:
print("Max nloop reached during mean-std normalization!",nloop)
break
# Ts
Ts=np.zeros(shape=[10,self.ngrid,3,3])
for i in range(self.ngrid):
sij = Sij[i, :, :]
oij = Oij[i, :, :]
Ts[0, i, :, :] = sij
Ts[1, i, :, :] = np.dot(sij, oij) - np.dot(oij, sij)
Ts[2, i, :, :] = np.dot(sij, sij) - 1./3.*np.eye(3)*np.trace(np.dot(sij, sij))
Ts[3, i, :, :] = np.dot(oij, oij) - 1./3.*np.eye(3)*np.trace(np.dot(oij, oij))
Ts[4, i, :, :] = np.dot(oij, np.dot(sij, sij)) - np.dot(np.dot(sij, sij), oij)
Ts[5, i, :, :] = np.dot(oij, np.dot(oij, sij)) \
+ np.dot(sij, np.dot(oij, oij)) \
- 2./3.*np.eye(3)*np.trace(np.dot(sij, np.dot(oij, oij)))
Ts[6, i, :, :] = np.dot(np.dot(oij, sij), np.dot(oij, oij)) - np.dot(np.dot(oij, oij), np.dot(sij, oij))
Ts[7, i, :, :] = np.dot(np.dot(sij, oij), np.dot(sij, sij)) - np.dot(np.dot(sij, sij), np.dot(oij, sij))
Ts[8, i, :, :] = np.dot(np.dot(oij, oij), np.dot(sij, sij)) \
+ np.dot(np.dot(sij, sij), np.dot(oij, oij)) \
- 2./3.*np.eye(3)*np.trace(np.dot(np.dot(sij, sij), np.dot(oij, oij)))
Ts[9, i, :, :] = np.dot(np.dot(oij, np.dot(sij, sij)), np.dot(oij, oij)) \
- np.dot(np.dot(oij, np.dot(oij, sij)), np.dot(sij, oij))
# Enforce zero trace for anisotropy
for j in range(9):
Ts[j, i, :, :] = Ts[j, i, :, :] - 1./3.*np.eye(3)*np.trace(Ts[j, i, :, :])
return(lambdas,Ts)
class ReynoldsStressTensor:
'''
Purpose: class of Reynolds stress tensor
Parameters
----------
tau_ij: 2D numpy array, float.
nrow = ngrid (number of grid)
ncol = nvar (e.g., uu,vv,ww,uv for 2D; uu,vv,ww,uv,uw,vw for 3D)
'''
def __init__(self, tau_ij=np.ones([1,6])):
try:
[ngrid, nvar] = tau_ij.shape
self.ngrid = ngrid
self.nvar = nvar
if nvar == 4: # 2D data with uu,vv,ww,uv
print('Initialize ReynoldsStressTensor using 2D data...')
tau_ij = np.concatenate((tau_ij, np.zeros(shape=[ngrid, 2])), axis=1)
elif nvar == 6: # 3D data with uu,vv,ww,uv,uw,vw
print('Initialize ReynoldsStressTensor using 3D data...')
else:
raise ValueError
except:
raise ValueError('Wrong data type: please use 2D numpy array with ncol = 4 or 6 for ReynoldsStressTensor')
self.Components = tau_ij
self.TKE = self.CalcTKE
self.AnisotropyEigenVal = self.CalcAnisotropyEigenVal
def __str__(self):
return("Reynolds stress tensors of %.i grid points."%(self.ngrid))
@property
def CalcTKE(self):
'''
Purpose: calculate turbulence kinetic energy
'''
k = np.sum(self.Components[:,0:3], axis=1)/2
return(k)
@property
def CalcAnisotropyTensor(self):
'''
Purpose: calculate turbulence anisotropy tensor
'''
k = self.TKE
k[k<=0] = 1e-9 # avoid division by zero
a_ij = self.Components/2/np.reshape(k, newshape=(-1,1))
a_ij[:,0:3] -= 1/3
return(a_ij)
@property
def CalcAnisotropyEigenVal(self):
'''
Purpose: calculate eigenvalues of turbulence anisotropy tensor
'''
a_ij = self.CalcAnisotropyTensor
EigenVals = []
EigenVal_flags = [] # flags used for debug
# loop over each grid point
for igrid in range(self.Components.shape[0]):
matrix = np.array([[a_ij[igrid,0],a_ij[igrid,3],a_ij[igrid,4]],
[a_ij[igrid,3],a_ij[igrid,1],a_ij[igrid,5]],
[a_ij[igrid,4],a_ij[igrid,5],a_ij[igrid,2]]])
lambdas = linalg.eigvals(matrix) # solve for eigenvalues, i.e., lambdas
lambdas_real = np.real(lambdas)
lambdas_real[::-1].sort() # descending order of eigenvalues
EigenVals.append(lambdas_real)
# check if eigenvalues are correct
if np.sum(np.imag(lambdas)) > 0.001:
print('Non-real eigenvalues detected! Please check data.')
print('Error occurs at index: '+str(igrid))
EigenVal_flags.append(-1)
if np.max(np.real(lambdas))>2/3 or np.min(np.real(lambdas))<-2/3:
EigenVal_flags.append(0)
print('Eigenvalues beyond range [-2/3,2/3] detected! Please check data.')
print('Error occurs at index: '+str(igrid))
else:
EigenVal_flags.append(1)
return(np.array(EigenVals))
def LumleyTriCoor(self):
'''
Purpose: calculate turbulence data coordinates in Lumley triangle
'''
EigenVals = self.AnisotropyEigenVal
PC2 = EigenVals[:,0]**2 + EigenVals[:,1]**2 + EigenVals[:,0]*EigenVals[:,1]
PC3 = -EigenVals[:,0]*EigenVals[:,1]*(EigenVals[:,0] + EigenVals[:,1])
PCs = np.concatenate((PC3.reshape([-1,1]), PC2.reshape([-1,1])), axis=1)
return(PCs)
def TurbTriCoor(self):
'''
Purpose: calculate turbulence data coordinates in turbulence triangle
'''
EigenVals = self.AnisotropyEigenVal
xi = cubic_root((-EigenVals[:,0]*EigenVals[:,1]*(EigenVals[:,0]+EigenVals[:,1]))/2)
eta = ((EigenVals[:,0]**2 + EigenVals[:,1]**2 + EigenVals[:,0]*EigenVals[:,1])/3)**0.5
corrs = np.concatenate((xi.reshape([-1,1]), eta.reshape([-1,1])), axis=1)
return(corrs)
def BaryTriCoor(self):
'''
Purpose: calculate turbulence data coordinates in barycentric map
'''
EigenVals = self.AnisotropyEigenVal
xB = EigenVals[:,0]-EigenVals[:,1]+(3*EigenVals[:,2]+1)/2
yB = (3*EigenVals[:,2]+1)*np.sqrt(3)/2
coors = np.concatenate((xB.reshape([-1,1]), yB.reshape([-1,1])), axis=1)
return(coors)
def AniRGB(self,c_off=0.65,c_exp=5):
'''
Purpose: calculate RGB values from eigenvalues of turbulence anisotropy tensor
'''
EigenVals = self.AnisotropyEigenVal
R = (EigenVals[:,0]-EigenVals[:,1]+c_off)**c_exp
G = (2*(EigenVals[:,1]-EigenVals[:,2])+c_off)**c_exp
B = (3*EigenVals[:,2]+1+c_off)**c_exp
RGB = np.concatenate((R.reshape([-1,1]), G.reshape([-1,1]), B.reshape([-1,1])), axis=1)
RGB[RGB>1] = 1
return(RGB)
# -------------------------------------------------------------------------
# sub-functions
def cubic_root(data):
'''
Purpose: Calculate cubic root and return the real root
Parameters
----------
data: numpy array, float.
Return
root: numpy array, float.
'''
data_shape=data.shape
data=data.flatten()
root=[]
for idx in range(data.shape[0]):
if data[idx] >= 0:
root.append(data[idx]**(1/3))
else:
root.append(-(-data[idx])**(1/3))
root=np.array(root)
root=root.reshape(data_shape)
return root
def calc_ReynoldsStressTensor(MeanGrad, EddyVisc, TKE=0, omega=1, method='Boussinesq'):
'''
Purpose: Calculate Reynolds stress tensor based on a constitutive relation
using velocity gradients, eddy viscosity and TKE
Parameters
----------
MeanGrad : MeanGradField Class.
EddyVisc : 1D numpy array, float. Kinematic eddy viscosity
TKE : 1D numpy array, float. Turbulence kinetic energy (default 0)
omega : 1D numpy array, float. Turbulence specific dissipation rate (default 1)
Method : string; 'Boussinesq'(default), 'QCR2000', 'QCR2013', 'QCR2013V', 'EARSM2005-2D'
Return
RST: ReynoldsStressTensor Class.
'''
# initialize parameters
ng = EddyVisc.shape[0]
RST = np.zeros(shape=[ng,6])
# Linear term
if method in ['Boussinesq','QCR2000','QCR2013','QCR2013V','EARSM2005-2D']:
# symmetric stress tensor
S_trace = MeanGrad.StrainRate[:,0]+MeanGrad.StrainRate[:,1]+MeanGrad.StrainRate[:,2]
S11 = MeanGrad.StrainRate[:,0]-S_trace/3
S22 = MeanGrad.StrainRate[:,1]-S_trace/3
S33 = MeanGrad.StrainRate[:,2]-S_trace/3
S12 = MeanGrad.StrainRate[:,3]
S13 = MeanGrad.StrainRate[:,4]
S23 = MeanGrad.StrainRate[:,5]
S21 = S12
S31 = S13
S32 = S23
# Linear Boussinesq part of Reynolds stress
RST[:,0] = -2*EddyVisc*S11
RST[:,1] = -2*EddyVisc*S22
RST[:,2] = -2*EddyVisc*S33
RST[:,3] = -2*EddyVisc*S12
RST[:,4] = -2*EddyVisc*S13
RST[:,5] = -2*EddyVisc*S23
# Quadratic terms
if method in ['QCR2000','QCR2013','QCR2013V']:
ccr1 = 0.3
# antisymmetric normalized rotation tensor
ugrad_mag = np.sqrt(np.sum((MeanGrad.VelGrad)**2, axis=1))
O11 = 0
O22 = 0
O33 = 0
O12 = 2*MeanGrad.Vorticity[:,0]/(ugrad_mag+1e-12)
O13 = 2*MeanGrad.Vorticity[:,1]/(ugrad_mag+1e-12)
O23 = 2*MeanGrad.Vorticity[:,2]/(ugrad_mag+1e-12)
O21 = -O12
O31 = -O13
O32 = -O23
# add quadratic terms
delta_RST = np.zeros(shape=[ng,6])
delta_RST[:,0] = -ccr1*(2*O12*RST[:,3]+2*O13*RST[:,4])
delta_RST[:,1] = -ccr1*(2*O21*RST[:,3]+2*O23*RST[:,5])
delta_RST[:,2] = -ccr1*(2*O31*RST[:,4]+2*O32*RST[:,5])
delta_RST[:,3] = -ccr1*(O12*RST[:,1]+O13*RST[:,5]+\
O21*RST[:,0]+O23*RST[:,4])
delta_RST[:,4] = -ccr1*(O12*RST[:,5]+O13*RST[:,2]+\
O31*RST[:,0]+O32*RST[:,3])
delta_RST[:,5] = -ccr1*(O21*RST[:,4]+O23*RST[:,2]+\
O31*RST[:,3]+O32*RST[:,1])
RST = RST+delta_RST
elif method in ['EARSM2005-2D']:
Neq = 81/20
Cdiff = 2.2
betastar = 0.09
# turbulence time scale
tt = 1/(omega+1e-12)/betastar
# symmetric normalized stress tensor
S11 = tt*S11
S22 = tt*S22
S33 = tt*S33
S12 = tt*S12
S13 = tt*S13
S23 = tt*S23
S21 = S12
S31 = S13
S32 = S23
II_S = S11*S11+S12*S21+S13*S31+\
S21*S12+S22*S22+S23*S32+\
S31*S13+S32*S23+S33*S33
# antisymmetric normalized rotation tensor
O11 = 0
O22 = 0
O33 = 0
O12 = 2*MeanGrad.Vorticity[:,0]*tt
O13 = 2*MeanGrad.Vorticity[:,1]*tt
O23 = 2*MeanGrad.Vorticity[:,2]*tt
O21 = -O12
O31 = -O13
O32 = -O23
II_O = O11*O11+O12*O21+O13*O31+\
O21*O12+O22*O22+O23*O32+\
O31*O13+O32*O23+O33*O33
# add quadratic terms
beta1eq = -6/5*Neq/(Neq**2-2*II_O)
A3p = 9/5+9/4*Cdiff*np.clip(1+beta1eq*II_S, 0, None)
P1 = (A3p**2/27+9/20*II_S-2/3*II_O)*A3p
P2 = P1**2-(A3p/9+9/10*II_S+2/3*II_O)**3
P2p_idx = P2>=0
P2n_idx = P2<0
Np = A3p/3+(P1+np.sqrt(np.abs(P2)))**(1/3)+np.sign(P1-np.sqrt(np.abs(P2)))*(np.abs(P1-np.sqrt(np.abs(P2))))**(1/3)
Nn = A3p/3+2*(P1**2+np.abs(P2))**(1/6)+np.cos(np.arccos(P1/(np.sqrt(P1**2+np.abs(P2))))/3)
N = np.zeros(shape=Np.shape)
N[P2p_idx] = Np[P2p_idx]
N[P2n_idx] = Nn[P2n_idx]
beta4_2D = -6/5/(N**2-2*II_O)
# add quadratic terms
delta_RST = np.zeros(shape=[ng,6])
delta_RST[:,0] = S11*O11-O11*S11+\
S12*O21-O12*S21+\
S13*O31-O13*S31
delta_RST[:,1] = S21*O12-O21*S12+\
S22*O22-O22*S22+\
S23*O32-O23*S32
delta_RST[:,2] = S31*O13-O31*S13+\
S32*O23-O32*S23+\
S33*O33-O33*S33
delta_RST[:,3] = S11*O12-O11*S12+\
S12*O22-O12*S22+\
S13*O32-O13*S32
delta_RST[:,4] = S11*O13-O11*S13+\
S12*O23-O12*S23+\
S13*O33-O13*S33
delta_RST[:,5] = S21*O13-O21*S13+\
S22*O23-O22*S23+\
S23*O33-O23*S33
delta_RST = delta_RST*np.reshape(TKE, (len(TKE), 1))*np.reshape(beta4_2D, (len(beta4_2D), 1))
RST = RST+delta_RST
# TKE diagonal term
if (method == 'QCR2013') and (np.sum(np.abs(TKE)) < 1e-9):
# mimic the TKE term using QCR2013
ccr2 = 2.5
S_mag = np.sqrt(S11**2+S12**2+S13**2+
S12**2+S22**2+S23**2+
S13**2+S23**2+S33**2)*np.sqrt(2)
RST[:,0] += ccr2*EddyVisc*S_mag
RST[:,1] += ccr2*EddyVisc*S_mag
RST[:,2] += ccr2*EddyVisc*S_mag
elif (method == 'QCR2013V') and (np.sum(np.abs(TKE)) < 1e-9):
# mimic the TKE term using QCR2013V
ccr2 = 2.5
W_mag = np.sqrt(O11**2+O12**2+O13**2+
O21**2+O22**2+O23**2+
O31**2+O32**2+O33**2)* \
(ugrad_mag+1e-12)/np.sqrt(2)
RST[:,0] += ccr2*EddyVisc*W_mag
RST[:,1] += ccr2*EddyVisc*W_mag
RST[:,2] += ccr2*EddyVisc*W_mag
else:
# use user-input TKE term
RST[:,0] += 2/3*TKE
RST[:,1] += 2/3*TKE
RST[:,2] += 2/3*TKE
return RST
def calc_EddyVisc_general(RST, MeanGrad, method='QCR2013V',inisol=None):
'''
Purpose: Calculate eddy viscosity based on constitutive relation using
Reynolds stress components and velocity gradients (general version
based on optimization algorithm; not numerically efficient)
Parameters
----------
RST: ReynoldsStressTensor Class.
MeanGrad: MeanGradField Class.
Method: string; 'Boussinesq', 'QCR2000', 'QCR2013', 'QCR2013V'(default), 'EARSM2005'
Return
EddyVisc: 1D numpy array, float. Calculated kinematic eddy viscosity
'''
from scipy.optimize import minimize
def func(nu_t_i,RST_i,MeanGrad_i,method):
Recon_RST_i = calc_ReynoldsStressTensor(MeanGrad_i, nu_t_i, TKE=RST_i.TKE,
omega=RST_i.TKE/np.clip(nu_t_i,1e-9,None), method=method)
weight=np.column_stack((np.ones((1, 3)), np.full((1, 3), 2)))
f = np.sum((Recon_RST_i-RST_i.Components)**2*weight)
return f
EddyVisc = []
ng = RST.ngrid
for i in range(ng):
if np.sum(RST.Components[i,0:3])<0.01:
EddyVisc.append(1e-6)
else:
RST_i = ReynoldsStressTensor(np.reshape(RST.Components[i,:], (1, 6)))
MeanGrad_i = MeanGradField(np.reshape(MeanGrad.VelGrad[i,:], (1, 9)))
try:
nu_t_ini_i = inisol[i]
except:
nu_t_ini_i = 1e-5
nu_t_i = minimize(func, nu_t_ini_i, method='L-BFGS-B', args=(RST_i, MeanGrad_i, method),
options={'maxiter': 1000, 'eps': 1e-6})
EddyVisc.append(nu_t_i['x'][0])
EddyVisc = np.array(EddyVisc)
return EddyVisc
def calc_EddyVisc(RST, MeanGrad, S_ref=100, method='QCR2013V'):
'''
Purpose: Calculate eddy viscosity based on constitutive relation using
Reynolds stress components and velocity gradients
Parameters
----------
RST: ReynoldsStressTensor Class.
MeanGrad: MeanGradField Class.
S_ref: real; limiter to avoid division by zero
Method: string; 'Boussinesq', 'QCR2000', 'QCR2013', 'QCR2013V'(default)
Return
EddyVisc: 1D numpy array, float. Calculated kinematic eddy viscosity
flag: indicator of regions affected by limiters.
0: no limiter; 1: S_ref applied; -1: EddyVisc=max{0,self} applied.
'''
# initialize parameters
ng = (RST.Components).shape[0]
uu = RST.Components[:,0]
vv = RST.Components[:,1]
ww = RST.Components[:,2]
uv = RST.Components[:,3]
uw = RST.Components[:,4]
vw = RST.Components[:,5]
dudx = MeanGrad.VelGrad[:,0]
dudy = MeanGrad.VelGrad[:,1]
dudz = MeanGrad.VelGrad[:,2]
dvdx = MeanGrad.VelGrad[:,3]
dvdy = MeanGrad.VelGrad[:,4]
dvdz = MeanGrad.VelGrad[:,5]
dwdx = MeanGrad.VelGrad[:,6]
dwdy = MeanGrad.VelGrad[:,7]
dwdz = MeanGrad.VelGrad[:,8]
flag = np.zeros(shape=(ng,1))
# calc eddy viscosity
if method=='Boussinesq':
# default
denominator = 2*(dudx**2+dvdy**2+dwdz**2+
2*(0.5*(dudy+dvdx))**2+
2*(0.5*(dudz+dwdx))**2+
2*(0.5*(dvdz+dwdy))**2-
1/3*(dudx+dvdy+dwdz)**2)
numerator = -(uu*dudx+vv*dvdy+ww*dwdz+
2*uv*0.5*(dudy+dvdx)+
2*uw*0.5*(dudz+dwdx)+
2*vw*0.5*(dvdz+dwdy)-
1/3*(uu+vv+ww)*(dudx+dvdy+dwdz))
# # the following ignores the uu, vv, ww accuracy, because one-equation model does not have k
# # and thus cannot predict uu, vv, ww; the absolute value of the Reynolds normal stress does
# # not matter much anyway, it is the dtau_ii/dx_j that matters.
# denominator = 2*(2*(0.5*(dudy+dvdx))**2+
# 2*(0.5*(dudz+dwdx))**2+
# 2*(0.5*(dvdz+dwdy))**2)
# numerator = -(2*uv*0.5*(dudy+dvdx)+
# 2*uw*0.5*(dudz+dwdx)+
# 2*vw*0.5*(dvdz+dwdy))
elif method in ['QCR2000','QCR2013','QCR2013V']:
ccr1 = 0.3
ccr2 = 2.5
ugrad_mag = np.sqrt(dudx**2+dudy**2+dudz**2+
dvdx**2+dvdy**2+dvdz**2+
dwdx**2+dwdy**2+dwdz**2)
# strain rate star
S_trace = dudx+dvdy+dwdz
S11 = dudx-S_trace/3
S22 = dvdy-S_trace/3
S33 = dwdz-S_trace/3
S12 = 0.5*(dudy+dvdx)
S13 = 0.5*(dudz+dwdx)
S23 = 0.5*(dvdz+dwdy)
S21 = S12
S31 = S13
S32 = S23
# antisymmetric normalized rotation tensor
O11 = 0
O22 = 0
O33 = 0
O12 = (dudy-dvdx)/(ugrad_mag+1e-12)
O13 = (dudz-dwdx)/(ugrad_mag+1e-12)
O23 = (dvdz-dwdy)/(ugrad_mag+1e-12)
O21 = -O12
O31 = -O13
O32 = -O23
if method=='QCR2000': # deprecated for 1-eqn model; for 2-eqn models replace k by the values predicted by the model
# the following uses k (default)
k = 0 # assume k=0 for SA models
denominator = 2*((S11-ccr1*(O11*S11+O11*S11+O12*S12+O12*S12+O13*S13+O13*S13))**2+\
(S12-ccr1*(O11*S21+O21*S11+O12*S22+O22*S12+O13*S23+O23*S13))**2+\
(S13-ccr1*(O11*S31+O31*S11+O12*S32+O32*S12+O13*S33+O33*S13))**2+\
(S21-ccr1*(O21*S11+O11*S21+O22*S12+O12*S22+O23*S13+O13*S23))**2+\
(S22-ccr1*(O21*S21+O21*S21+O22*S22+O22*S22+O23*S23+O23*S23))**2+\
(S23-ccr1*(O21*S31+O31*S21+O22*S32+O32*S22+O23*S33+O33*S23))**2+\
(S31-ccr1*(O31*S11+O11*S31+O32*S12+O12*S32+O33*S13+O13*S33))**2+\
(S32-ccr1*(O31*S21+O21*S31+O32*S22+O22*S32+O33*S23+O23*S33))**2+\
(S33-ccr1*(O31*S31+O31*S31+O32*S32+O32*S32+O33*S33+O33*S33))**2)
numerator = (S11-ccr1*(O11*S11+O11*S11+O12*S12+O12*S12+O13*S13+O13*S13))*(-uu+k*2/3)+\
(S12-ccr1*(O11*S21+O21*S11+O12*S22+O22*S12+O13*S23+O23*S13))*(-uv)+\
(S13-ccr1*(O11*S31+O31*S11+O12*S32+O32*S12+O13*S33+O33*S13))*(-uw)+\
(S21-ccr1*(O21*S11+O11*S21+O22*S12+O12*S22+O23*S13+O13*S23))*(-uv)+\
(S22-ccr1*(O21*S21+O21*S21+O22*S22+O22*S22+O23*S23+O23*S23))*(-vv+k*2/3)+\
(S23-ccr1*(O21*S31+O31*S21+O22*S32+O32*S22+O23*S33+O33*S23))*(-vw)+\
(S31-ccr1*(O31*S11+O11*S31+O32*S12+O12*S32+O33*S13+O13*S33))*(-uw)+\
(S32-ccr1*(O31*S21+O21*S31+O32*S22+O22*S32+O33*S23+O23*S33))*(-vw)+\
(S33-ccr1*(O31*S31+O31*S31+O32*S32+O32*S32+O33*S33+O33*S33))*(-ww+k*2/3)
# # the following uses hi-fi TKE instead of k predicted by RANS
# denominator = 2*((S11-ccr1*(O11*S11+O11*S11+O12*S12+O12*S12+O13*S13+O13*S13))**2+\
# (S12-ccr1*(O11*S21+O21*S11+O12*S22+O22*S12+O13*S23+O23*S13))**2+\
# (S13-ccr1*(O11*S31+O31*S11+O12*S32+O32*S12+O13*S33+O33*S13))**2+\
# (S21-ccr1*(O21*S11+O11*S21+O22*S12+O12*S22+O23*S13+O13*S23))**2+\
# (S22-ccr1*(O21*S21+O21*S21+O22*S22+O22*S22+O23*S23+O23*S23))**2+\
# (S23-ccr1*(O21*S31+O31*S21+O22*S32+O32*S22+O23*S33+O33*S23))**2+\
# (S31-ccr1*(O31*S11+O11*S31+O32*S12+O12*S32+O33*S13+O13*S33))**2+\
# (S32-ccr1*(O31*S21+O21*S31+O32*S22+O22*S32+O33*S23+O23*S33))**2+\
# (S33-ccr1*(O31*S31+O31*S31+O32*S32+O32*S32+O33*S33+O33*S33))**2)
# numerator = (S11-ccr1*(O11*S11+O11*S11+O12*S12+O12*S12+O13*S13+O13*S13))*(-uu+(uu+vv+ww)/3)+\
# (S12-ccr1*(O11*S21+O21*S11+O12*S22+O22*S12+O13*S23+O23*S13))*(-uv)+\
# (S13-ccr1*(O11*S31+O31*S11+O12*S32+O32*S12+O13*S33+O33*S13))*(-uw)+\
# (S21-ccr1*(O21*S11+O11*S21+O22*S12+O12*S22+O23*S13+O13*S23))*(-uv)+\
# (S22-ccr1*(O21*S21+O21*S21+O22*S22+O22*S22+O23*S23+O23*S23))*(-vv+(uu+vv+ww)/3)+\
# (S23-ccr1*(O21*S31+O31*S21+O22*S32+O32*S22+O23*S33+O33*S23))*(-vw)+\
# (S31-ccr1*(O31*S11+O11*S31+O32*S12+O12*S32+O33*S13+O13*S33))*(-uw)+\
# (S32-ccr1*(O31*S21+O21*S31+O32*S22+O22*S32+O33*S23+O23*S33))*(-vw)+\
# (S33-ccr1*(O31*S31+O31*S31+O32*S32+O32*S32+O33*S33+O33*S33))*(-ww+(uu+vv+ww)/3)
# # the following ignores the uu, vv, ww accuracy, because one-equation model does not have k
# # and thus cannot predict uu, vv, ww; the absolute value of the Reynolds normal stress does
# # not matter much anyway, it is the dtau_ii/dx_j that matters.
# denominator = 2*((S12-ccr1*(O11*S21+O21*S11+O12*S22+O22*S12+O13*S23+O23*S13))**2+\
# (S13-ccr1*(O11*S31+O31*S11+O12*S32+O32*S12+O13*S33+O33*S13))**2+\
# (S21-ccr1*(O21*S11+O11*S21+O22*S12+O12*S22+O23*S13+O13*S23))**2+\
# (S23-ccr1*(O21*S31+O31*S21+O22*S32+O32*S22+O23*S33+O33*S23))**2+\
# (S31-ccr1*(O31*S11+O11*S31+O32*S12+O12*S32+O33*S13+O13*S33))**2+\
# (S32-ccr1*(O31*S21+O21*S31+O32*S22+O22*S32+O33*S23+O23*S33))**2)
# numerator = (S12-ccr1*(O11*S21+O21*S11+O12*S22+O22*S12+O13*S23+O23*S13))*(-uv)+\
# (S13-ccr1*(O11*S31+O31*S11+O12*S32+O32*S12+O13*S33+O33*S13))*(-uw)+\
# (S21-ccr1*(O21*S11+O11*S21+O22*S12+O12*S22+O23*S13+O13*S23))*(-uv)+\
# (S23-ccr1*(O21*S31+O31*S21+O22*S32+O32*S22+O23*S33+O33*S23))*(-vw)+\
# (S31-ccr1*(O31*S11+O11*S31+O32*S12+O12*S32+O33*S13+O13*S33))*(-uw)+\
# (S32-ccr1*(O31*S21+O21*S31+O32*S22+O22*S32+O33*S23+O23*S33))*(-vw)
if method=='QCR2013': # deprecated for 2-eqn model; for 1-eqn model prefer to use 2013V
pseudo_k = np.sqrt(2*(S11**2+S12**2+S13**2+S21**2+S22**2+S23**2+S31**2+S32**2+S33**2))
denominator = 2*((S11-ccr1*(O11*S11+O11*S11+O12*S12+O12*S12+O13*S13+O13*S13))**2+\
(S12-ccr1*(O11*S21+O21*S11+O12*S22+O22*S12+O13*S23+O23*S13))**2+\
(S13-ccr1*(O11*S31+O31*S11+O12*S32+O32*S12+O13*S33+O33*S13))**2+\
(S21-ccr1*(O21*S11+O11*S21+O22*S12+O12*S22+O23*S13+O13*S23))**2+\
(S22-ccr1*(O21*S21+O21*S21+O22*S22+O22*S22+O23*S23+O23*S23)-ccr2/2*pseudo_k)**2+\
(S23-ccr1*(O21*S31+O31*S21+O22*S32+O32*S22+O23*S33+O33*S23))**2+\
(S31-ccr1*(O31*S11+O11*S31+O32*S12+O12*S32+O33*S13+O13*S33))**2+\
(S32-ccr1*(O31*S21+O21*S31+O32*S22+O22*S32+O33*S23+O23*S33))**2+\
(S33-ccr1*(O31*S31+O31*S31+O32*S32+O32*S32+O33*S33+O33*S33)-ccr2/2*pseudo_k)**2)
numerator = (S11-ccr1*(O11*S11+O11*S11+O12*S12+O12*S12+O13*S13+O13*S13)-ccr2/2*pseudo_k)*(-uu)+\
(S12-ccr1*(O11*S21+O21*S11+O12*S22+O22*S12+O13*S23+O23*S13))*(-uv)+\
(S13-ccr1*(O11*S31+O31*S11+O12*S32+O32*S12+O13*S33+O33*S13))*(-uw)+\
(S21-ccr1*(O21*S11+O11*S21+O22*S12+O12*S22+O23*S13+O13*S23))*(-uv)+\
(S22-ccr1*(O21*S21+O21*S21+O22*S22+O22*S22+O23*S23+O23*S23)-ccr2/2*pseudo_k)*(-vv)+\
(S23-ccr1*(O21*S31+O31*S21+O22*S32+O32*S22+O23*S33+O33*S23))*(-vw)+\
(S31-ccr1*(O31*S11+O11*S31+O32*S12+O12*S32+O33*S13+O13*S33))*(-uw)+\
(S32-ccr1*(O31*S21+O21*S31+O32*S22+O22*S32+O33*S23+O23*S33))*(-vw)+\
(S33-ccr1*(O31*S31+O31*S31+O32*S32+O32*S32+O33*S33+O33*S33)-ccr2/2*pseudo_k)*(-ww)
if method=='QCR2013V': # deprecated for 2-eqn model
pseudo_k = np.sqrt(2*(O11**2+O12**2+O13**2+O21**2+O22**2+O23**2+O31**2+O32**2+O33**2))*ugrad_mag/2
denominator = 2*((S11-ccr1*(O11*S11+O11*S11+O12*S12+O12*S12+O13*S13+O13*S13))**2+\
(S12-ccr1*(O11*S21+O21*S11+O12*S22+O22*S12+O13*S23+O23*S13))**2+\
(S13-ccr1*(O11*S31+O31*S11+O12*S32+O32*S12+O13*S33+O33*S13))**2+\
(S21-ccr1*(O21*S11+O11*S21+O22*S12+O12*S22+O23*S13+O13*S23))**2+\
(S22-ccr1*(O21*S21+O21*S21+O22*S22+O22*S22+O23*S23+O23*S23)-ccr2/2*pseudo_k)**2+\
(S23-ccr1*(O21*S31+O31*S21+O22*S32+O32*S22+O23*S33+O33*S23))**2+\
(S31-ccr1*(O31*S11+O11*S31+O32*S12+O12*S32+O33*S13+O13*S33))**2+\
(S32-ccr1*(O31*S21+O21*S31+O32*S22+O22*S32+O33*S23+O23*S33))**2+\
(S33-ccr1*(O31*S31+O31*S31+O32*S32+O32*S32+O33*S33+O33*S33)-ccr2/2*pseudo_k)**2)
numerator = (S11-ccr1*(O11*S11+O11*S11+O12*S12+O12*S12+O13*S13+O13*S13)-ccr2/2*pseudo_k)*(-uu)+\
(S12-ccr1*(O11*S21+O21*S11+O12*S22+O22*S12+O13*S23+O23*S13))*(-uv)+\
(S13-ccr1*(O11*S31+O31*S11+O12*S32+O32*S12+O13*S33+O33*S13))*(-uw)+\
(S21-ccr1*(O21*S11+O11*S21+O22*S12+O12*S22+O23*S13+O13*S23))*(-uv)+\
(S22-ccr1*(O21*S21+O21*S21+O22*S22+O22*S22+O23*S23+O23*S23)-ccr2/2*pseudo_k)*(-vv)+\
(S23-ccr1*(O21*S31+O31*S21+O22*S32+O32*S22+O23*S33+O33*S23))*(-vw)+\
(S31-ccr1*(O31*S11+O11*S31+O32*S12+O12*S32+O33*S13+O13*S33))*(-uw)+\
(S32-ccr1*(O31*S21+O21*S31+O32*S22+O22*S32+O33*S23+O23*S33))*(-vw)+\
(S33-ccr1*(O31*S31+O31*S31+O32*S32+O32*S32+O33*S33+O33*S33)-ccr2/2*pseudo_k)*(-ww)
else:
raise ValueError("User input 'method' is not supported")
# limit |denominator|>= S_ref**2
S_ref2 = S_ref**2
idx_sref_neg = np.where((denominator>=-S_ref2)&(denominator<= 0))
idx_sref_pos = np.where((denominator<= S_ref2)&(denominator>= 0))
idx_sref = np.concatenate((idx_sref_neg[0].flatten(),idx_sref_pos[0].flatten()))
if idx_sref != []:
flag[idx_sref,0]=1
# flag[idx_sref,0]=1-np.abs(denominator[idx_sref])/S_ref2
denominator[idx_sref_neg]=-S_ref2
denominator[idx_sref_pos]= S_ref2
# calc eddy viscosity
EddyVisc = numerator/denominator
# limit EddyVisc >= 0
idx_negEddyVisc = np.where(EddyVisc<0)[0]
EddyVisc[idx_negEddyVisc] = 0
if idx_negEddyVisc != []:
for idx in idx_negEddyVisc:
if flag[idx,0] == 0:
flag[idx,0] = 2
elif flag[idx,0] == 1:
flag[idx,0] = 3
return EddyVisc, flag
def calc_Nutilde(nu_t,nu_l):
'''
Purpose: Calculate nu_tilde, the variable of the SA transport equation, from
eddy viscosity nu_t and laminar viscosity nu_l
Parameters
----------
nu_t: 1D numpy array, float.
nu_l: 1D numpy array, float.
Return
nu_tilde: 1D numpy array, float.
'''
from scipy.optimize import fsolve
def func(nu_tilde):
xx = nu_tilde/nu_li
f = xx**4/(xx**3+7.1**3) - nu_ti/nu_li
return f
nu_tilde = []
for i in range(nu_t.shape[0]):
nu_ti = nu_t[i]
nu_li = nu_l[i]
nu_tilde.append(fsolve(func, nu_ti))
return(np.array(nu_tilde)[:,0])
def plot_Lumley_tri(method='w/o data', coors=None):
'''
Purpose: plot Lumley triangle template
Parameters
----------
method : string; w/o data (default) for plotting frame only; else for plotting data
coors : 2D numpy array, float; coordinates calculated from ReynoldsStressTensor.LumleyTriCoor
nrow = ngrid (number of grid)
ncol = 2; Lumley triangle coordinates PC3 and PC2
Returns
-------
fig : matplotlib figure
'''
fig = plt.figure(figsize=(6,6))
# plot data
if method != 'w/o data':
try:
plt.scatter(coors[:,0], coors[:,1], zorder=0)
except:
raise ValueError("User input 'coors' is not 2D numpy array")
# figure format
plt.xlabel('III')
plt.ylabel('II')
plt.axis([-0.02,0.08,-0.05,0.4])
plt.yticks(np.linspace(0,0.4,5))
# triangle bounds
lam_1s = [2/3, 1/6, 0]
lam_2s = [-1/3, 1/6, 0]
for iedge in range(3):
lam_1=np.linspace(lam_1s[iedge],lam_1s[(iedge+1)%3],100)
lam_2=np.linspace(lam_2s[iedge],lam_2s[(iedge+1)%3],100)
PC2_temp=lam_1**2+lam_2**2+lam_1*lam_2
PC3_temp=-lam_1*lam_2*(lam_1+lam_2)
plt.plot(PC3_temp,PC2_temp,color='grey')
# texts
plt.text(-0.0027,-0.025,'3C',fontsize=16)
plt.text(-0.018,0.075,'2C',fontsize=16)
plt.text(0.07,0.34,'1C',fontsize=16)
return fig
def plot_turb_tri(method='w/o data', coors=None):
'''
Purpose: plot turbulence triangle
Parameters
----------
method : string; w/o data (default) for plotting frame only; else for plotting data
coors : 2D numpy array, float; coordinates calculated from ReynoldsStressTensor.TurbTriCoor
nrow = ngrid (number of grid)
ncol = 2; turbulence triangle coordinates xi and eta
Returns
-------
fig : matplotlib figure
'''
fig = plt.figure(figsize=(6,6))
# plot data
if method != 'w/o data':
try:
plt.scatter(coors[:,0], coors[:,1], zorder=0)
except:
raise ValueError("User input 'coors' is not 2D numpy array")
# figure format
plt.xlabel(r'$\xi$')
plt.ylabel(r'$\eta$')
plt.axis([-0.2,0.4,-0.03,0.35])