Peter A. Cavallo
8. Current Research Needs
While considerable research, development, and validation has been invested in the CRISP CFD code and it has had notable success for steady state problems, further work is needed to bring the error quantification tools into other problem classes. The temporal evolution of spatial errors remains an area of challenging research. Transient flows, particularly large eddy simulations (LES), may involve assessment of mean and fluctuating quantities. A natural extension of this problem class is to account for problems with moving or deforming boundaries. CFD is increasingly being applied to problems involving more complex physical phenomena as well. The prediction of sprays, particle-laden flows and free surface flows require additional submodels and/or transport equations, coupled to the conservation equations for the carrier phase in the case of dispersed multiphase flows. Error prediction for such problems would be beneficial in assessing the validity of new physical models by isolating discretization errors. With these advances, solution verification could be applied with even more complex test cases in validation databases such as CRAVE [67].
x/D
z/D
0 5 10
1 2 3 4 5 6 7
Original Grid First Adaptation Second Adaptation Richardson Extrapolation Experimental Data
x/D
GCI,%
0 5 10
0 5 10 15
a) Extrapolation of temperature trajectory b) Grid Convergence Index (GCI)
a) In the symmetry plane b) At selected streamwise stations
8.1. Errors in Chemical Kinetics
It can readily be seen from prior discussion that the accuracy of the chemical source term ultimately depends on the accuracy of the local species mass fractions and mixture temperature predictions. Numerical diffusion will adversely affect species production/destruction, as the local species concentrations and/or temperatures on a given grid may be lower than what is obtained in the fine grid limit. In the extreme, a coarse grid may prevent the formation of a flame entirely.
Consider the source term defined in Eqn. 8.
D! !
( )
Q −D!( )
Q!h (33)The source term D! ! Qh
( )
is evaluated on the given grid of mesh width h, using the actual solution obtained on that grid. The term D! !( )
Q represents the exact source term that would reflect the exact solution of the governing equations for the chemically reacting system. In essence, D! !( )
Q would be the source term for a grid-converged solution. Expressed in terms of the dependent variables, it becomes:D! !
( )
Q = f y(
iexact,Texact)
D!( )
Q!h = f y(
ih,Th)
(34)While the exact solution is not typically known, it may be approximated by applying the error predicted by the current solution of the ETE.
! D !
( )
Q = f y(
ih+εyi,Th+εT)
(35)Thus by evaluating the chemical source term using the solution and the predicted errors, we obtain a new residual term for the ETE, which represents error in the chemical source terms.
R!CHEM = ! D !
( )
Q −D!( )
Q!h( )
dV∫∫∫
(36)∂
∂t
ε!dV +
∫∫ ( (
A( )
Q!h ε!)
⋅ ˆn)
dA =∫∫ (
G !ε!( )
⋅ ˆn)
dA−R!INV∫∫∫
−R!TURB−R!CHEM (37)8.2. Errors In Transport (Scalar Fluctuations)
In our current implementation of the error transport method, turbulent Prandtl and Schmidt numbers are assumed to be constant, and as such the predicted errors in turbulent thermal conductivity and mass diffusivity, seen in Eqns. 18 and 19, are solely a function of the predicted error in eddy viscosity, as in Eqn. 23. A natural extension of this formulation is to expand the errors in these quantities to reflect a variable Prt and Sct field, as provided by a scalar fluctuation model (SFM). To accomplish this, the ETE would be modified to include error in the energy and mixture fraction variance equations of the SFM, from which one would obtain error in the conserved variables ρ , ke ρε , e ρkf, and ρε . f
Following the error propagation approach discussed previously, the error in turbulent Prandtl and
It follows that the errors in thermal conductivity and mass diffusivity originally defined in Eqn.
23 will now include additional terms:
Pr extension would be to produce a more accurate prediction of the error in temperature and species distribution in the domain.
8.3. Multiphase Flows
CFD is increasingly being applied to problems involving more complex physical phenomena such as gas/liquid or gas/solid flows. The prediction of sprays, particle-laden flows, and free surface flows require additional submodels and/or transport equations, coupled to the conservation equations for the carrier phase in the case of dispersed multiphase flows. Error prediction for such problems would be beneficial in assessing the validity of new physical models by isolating mesh-induced errors. Potential extensions to the error transport method include treatments for the source terms that provide for mass, momentum, and energy transfer between the carrier and dispersed phases, or treatments for assessing errors at a gas/liquid interface.
8.4. Transient Applications
The temporal evolution of spatial errors remains an area of challenging research. Research to date has not attempted to address temporal discretization errors, which may have a notable effect. At issue is how to separate errors due to the time-stepping scheme from errors caused by the mesh discretization when examining the spatial variation in the CFD solution. Transient flows, particularly LES, may involve an assessment of mean and fluctuating quantities, such as root mean square (RMS) pressure or velocity. In the future, the function library could include options for transient cases.
A natural extension of this problem class is to account for problems with moving or deforming boundaries. The additional aspect from a numerical perspective is the geometric conservation law (GCL). In problems involving moving meshes, the inviscid flux contains an additional term to account for the volume swept out by control volume faces in a given time step. Errors in this flux would certainly be an important element of error quantification for store release predictions, control surface deflections, missile staging, and similar applications.