Document
Numerical CFD S imulation and Test C orrelation in a
Flight P roject E nvironment
* ** ***
K. K. Gupta , S. F. Lung and A. H. Ibrahim
* NASA Armstrong Flight Research Center, Edwards, CA ** Jacobs Technology Inc., Edwards, CA *** Norfolk State University, Norfolk, VA
NASA Armstrong Flight Research Center
Outline
INTRODUCTION
FINITE ELEMENT FORMULATION
FINITE ELEMENT SOLUTION PROCEDURE
NUMERICAL EXAMPLE
RESULTS COMPARISON
CONCLUSIONS
Introduction
Develop in - house CFD codes to support NASA Armstrong flight research projects
Hyper X/X - 43 project Adaptive Compliant Trailing Edge (ACTE) project
Validate the in - house CFD codes
Compare with other commercially available CFD codes Compare with wind tunnel and flight test data
Summary of Flow Equations
Finite Element Formulation
Navier - Stokes equation:
𝜕 𝒗 𝜕 𝒇 𝜕 𝒈
𝒊 𝒊
+ + = 𝑓 𝑖 = 1 , 2 , 3
𝑏
𝜕𝑡 𝜕 𝑥 𝜕 𝑥
𝑖 𝑖
where
𝑇
𝜌 𝜌 𝑢 𝜌𝐸
𝒗 = , 𝑗 = 1 , 2 , 3
𝑗 𝑇
𝜌 𝑢 ( 𝜌 𝑢 𝑢 + 𝑝 𝛿 ) 𝑢 ( 𝑝 + 𝜌𝐸 )
𝑓 = , 𝑗 = 1 , 2 , 3
𝑗 𝑖 𝑗 𝑖𝑗 𝑗 𝑗 𝑇
𝜕𝑇
0 𝜎 𝑢 𝜎 + 𝑘
𝐸 =
𝑗 𝑖𝑗 𝑖 𝑖𝑗
𝜕 𝑥
𝑗 𝑇
0 𝑓 𝑢 𝑓
𝑓 =
𝑏 𝑖 𝑏 𝑏 𝑖 𝑖
𝜕 𝑢 𝜕 𝑢 2 𝜕 𝑢
𝑗 𝑖 𝑙
𝜎 = 𝜇 + − 𝛿 𝑙 = 1 , 2 , 3
𝑖𝑗 𝑖𝑗
𝜕 𝑥 𝜕 𝑥 3 𝜕 𝑥
𝑗 𝑖 𝑙 Nondimensionalized for numerical calculation, 𝒈 becomes: 𝒋
𝑔 = 0 𝜎 𝑢 𝜎 − 𝑞
𝑗 𝑖𝑗 𝑖 𝑖𝑗 𝑗
And
𝜇 𝜕 𝑢 𝜕 𝑢 2 𝜕 𝑢
𝑗 𝑖 𝑙
𝜎 = + − 𝛿
𝑖𝑗 𝑖𝑗
𝑅𝑒 𝜕 𝑥 𝜕 𝑥 3 𝜕 𝑥
𝑗 𝑖 𝑙
1 𝜕𝑇
𝑞 =
𝑗
𝑅𝑒𝑃𝑟 𝜕 𝑥
𝑗
Finite Element Formulation (Cont.)
Taylor’s expansion of the solution 𝒗 𝑥 , 𝑡 in the time domain:
𝜕 𝒇 𝜕 𝒈
𝒊 𝒊
∆ 𝒗 = − ∆ 𝑡 +
𝜕 𝑥 𝜕 𝑥
𝑖 𝑖 ( 𝑡 ) Applying Galerkin’s spatial idealization 𝒗 = 𝒂 𝒗 , the flow equation can be expressed as
𝜕 𝑢
𝑖
𝑴 ∆ 𝒗 = − ∆ 𝑡 𝑴 + 𝑲 𝒗 − ∆ 𝑡 𝒇 + 𝒇 + ∆ 𝑡 𝑹 + ∆ 𝑡 𝑲 + 𝒇
1 2 𝜎 𝜎
𝜕 𝑥
𝑖 A novel two - step solution procedure is adopted for the flow equation, the inviscid solution being augmented with the viscous term and stabilized with artificial dissipation terms . Assuming
∆ 𝒗 = 𝒗 − 𝒗
𝒏 + 𝟏 𝒏
Then
∆ 𝑡 ∆ 𝑡 ∆ 𝑡 ∆ 𝑡
1 + 𝑐 𝑴 + 𝑲 𝒗 = 1 − 𝑐 𝑴 − 𝑲 𝒗 + ∆ 𝑡 𝑹
𝑛 + 1 𝑛
2 2 2 2
Or
𝑴 𝒗 = 𝑴 𝒗 + ∆ 𝑡 𝑹
+ 𝑛 + 1 − 𝑛
Let
𝑴 = 𝑫 + 𝑴′
+ + + The viscous stress tensor and heat flux may then be solved as follows:
Finite Element Solution Procedure
Step 1: Form
𝑫 𝒗 = 𝑴 𝒗 − 𝑴′ 𝒗 + ∆ 𝑡 𝑹
+ 𝑛 + 1 − 𝑛 + 𝑛 + 1
Step 2: Solve 𝒗 iteratively
𝑛 + 1 ( 𝒊 + 𝟏 ) ( 𝒊 ) ( − 𝟏 ) ′
𝝊 = [ 𝑫 ] { [ 𝑴 ] 𝝊 − [ 𝑴 ] 𝝊 + ∆ 𝑡 ( 𝑹 + 𝑹 + 𝑲 + 𝒇 ) }
+ − 𝒏 + 𝝈 𝝈 𝒏 + 𝟏 𝒏 + 𝟏 𝒊 + 𝟏 𝒊 Step 3: If 𝒗 ≠ EPS1 𝒗 go to Step 2.
𝑛 + 1 𝑛 + 1 𝒊 + 𝟏 𝒊 Step 4: If 𝒗 ≠ EPS2 𝒗 go to Step 1.
𝑛 + 1 𝑛 + 1 Step 5: Repeat Steps 1 to 4 NITER times until desired convergence is achieved, that is until 𝒗 ≈ 𝑛 + 1 𝒗 ; EPS1 and EPS2 are suitable convergence criteria factors, specified by the users .
𝑛 Note: The iterative process in Step 2 requires a small number of steps, usually 1, and achieves a stable, convergent solution. In regions of high pressure gradients, artificial dissipation term is applied to prevent oscillations near discontinuities. This is implemented by incorporating pressure - switched diffusion coefficients as appropriate.
This procedure is adopted in the STARS - CFDSOL code that enables effective solution of the
Naviar - Stokes equation in most flight regimes
Numerical Example
Gulfstream III (GIII) Wing finite element discretization
15,538 grid 31,072 elements on wing surface 226,444 grid 1,256,879 million elements for the solution domain Flight condition: Mach 0.701, Angle of attack 3.92
Pressure Distributions
-1.5 STARS (MG) Wind Tunnel Flight Test -1 USM3D STARS (CFDSOL) STARCCM FUN3D -0.5 Fluent TRANAIR 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Pressure Distributions
-1.5 STARS (MG) Wind Tunnel Flight Test -1 USM3D STARS (CFDSOL) STARCCM FUN3D -0.5 FLUENT TRANAIR 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Pressure Distributions
-2 STARS (MG) Wind Tunnel Flight Test -1.5 USM3D STARS (CFDSOL) STARCCM -1 FUN3D FLUENT TRANAIR -0.5 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
C distribution on wing surface
p
C at station 145
p
C at station 230
p
C at station 385
p
CFD Codes Comparison
CFD Solver Flow Equation Platform No. of CPU Total CPU time Grid Size Note number of processers is an 7.2M RANS, finite 6hrs., 40min (533 estimate, and the polyhedra/prismatic STARCCM ++ volume, K - omega Cluster ~80 CPU hours) - 3000 time is an for half model without iterations estimate for that SST turbulence T - tail number of processors 8.12 M RANS, Finite 16 hrs., (512 CPU tetrahedral/prismatic Time is Fluent Volume, K - Omega Cluster 32 hours) for half model without approximate SST Turbulence T - tail 2.8 hrs. , ( 100 Dell M620 8GB 1 Intel Core i7 1.2 M Tetrahedrons for Euler, finite STARS (MG) steps, 25 inner Ram, 64 bit @2.67 GHz wing only element cycles) STARS Full N - S, finite Dell M620 8GB 1 Intel Core i7 13.8 hrs., (10000 2.8 M Tetrahedrons for (CFDSOL) element Ram, 64 bit @2.67 GHz steps) wing only 1.9 M cells for half Full N - S, finite USM3D Mac 64 bit 2 CPUs 16 hrs.
volume model without T - tail RANS, Finite 20.4 M nodes for half FUN3D Volume with Cluster 196 6 hrs.
model without T - tail turbulence model Full potential + Linux 1.7M cells for full TRANAIR++ viscosity 1 CPU 2hrs., 28min Workstation model (boundary layer)
Concluding Remarks
This paper presents detailed comparison of solutions of the GIII aircraft wing obtained by a number of commercially available CFD codes as well as two NASA AFRC in - house codes that use a finite element discretization employing unstructured grids.
Finite element formulation of the novel in - house CFD code (CFDSOL) is also presented.
Each of the codes shows reasonable correlation with flight and wing tunnel test data; CFDSOL and MG codes appears to be rather close to the two test results.
Use of a s ingle CPU to derive solutions testifies to their cost effectiveness.
CFDSOL Flow Chart
r egrid aero mesh Calculate Aeroelastic - Acoustic, SPL Acoustic wave frequencies