1. Fluid Flow Regularization of Navier-Stokes Equations
Elena Nikonova, Dr. Monika Neda
University of Nevada Las Vegas 2011
Introduction
Future Directions
Benchmark Problem
The Mesh
Fluid dynamics is the study of the fluid motion, such as water or air. Fluid Dynamics is used in prediction of weather prediction, (climate modeling, hurricane/tsunami warning), chemical engineering (refinement, mixing processes, fluid transportation such as oil flow in pipe lines), the aeronautics industry, oceanography, and astronomy.
Computational Fluid Dynamics (CFD) puts theory into practice and tries to simulate fluid flow with a computer.
We find numerical solutions to the Time Relaxation equations using the Finite Element Method. In order to reduce the approximation surface we need to introduce triangulation. We are considering the approximate surface to be globally continuous plane over each triangle. The mesh is a collection of these triangle faces as well as vertices and edges. The finer the mesh , the more accurate solution we obtain. We use a mesh to divide the domain into many triangular regions (“elements”):
The mesh shown is a coarse mesh.
Time Relaxation Model has an interesting parameter Χ Due to the fact that Χ causes different responses of the flow, it is natural to explore how exactly can the flow be affected by altering this parameter. This study is called a sensitivity analysis with respect to Χ.
Time relaxation model could also be tested on other benchmark problems such as the flow over a step and mixed layer problem.
Also, the simulations of the flow could be extended to 3 dimensions.
We use as a benchmark the problem of two-dimensional flow over a step. The following plots show the behavior of the problem applied to NSE with Re=600 on a fine mesh:
Table 1. NSE simulations applied to the step problem.
Now we compare it to the simulations using TRM done on the coarse mesh with parameters Re=600 and δ=1.5 with N=0 and N=1.
Table 1. TRM with order of deconvolution N=0.
Table 2. TRM with order of deconvolution N=1.
From the simulations we can see that the model performs well on the benchmark problem as it gives us the similar result to the NSE, but on the coarse mesh, and thus requires less computational time.
Main Concept
The basis for all fluid flow models are time dependent Navier Stokes equations (NSE):
In order to simulate the flow we need to know the shape of the domain, which is represented by Ω.
T represents final time, ν represents kinematic viscosity of the fluid and f represents the external force. Also, u is velocity at which the fluid moves and p is the pressure of the fluid. Fluid flow can be predicted by solving these equations for u and p at each time step.
In order to obtain a meaningful solution one needs to work with a fine mesh, which is beyond the scope of most computers to calculate. For this reason fluid flow models are developed, that can efficiently simulate the flow on coarse meshes.
In this this study we introduce Time Relaxation Model (TRM) developed by Adams, Stoltz and Kleiser, which adds an additional stabilization term to the Navier Stokes Equations:
represents filtered velocity, while represents deconvolution operator. What that basically means is that we want to filter out the small eddies. For different of order of deconvoltion N = 0, 1, 2 we have the following:
In order to do the filtering we need to know the filter radius, which is given by δ. Typically δ is taken of order of the mesh. The filtering operation for our model is given below:
The higher the order of deconvolution, the more accurate solution we obtain, however it becomes computationally expensive and time consuming as more filtering has to be done.
Acknowledgements
Errors
Our thanks go out to our predecessors, and most particularly to Drs. Ervin, Layton and Neda for providing detailed material on this benchmark problem. Thanks also Adams, Stoltz and Kleiser for developing the Time Relaxation Model.
Special thanks to the National Supercomputing Center for Energy and the Environment (NSCEE) and Center for Applied Mathematics and Statistics (CAMS) for lending much-needed computational power.
Now we want to see the errors with respect to the results for the Chorin problem. All the computations have been carried out with Re = 100, and Taylor-Hood finite elements are used (i.e. k=2, s=1). The tables below show errors and rates of convergence for the velocity and pressure for parameter of χ =
The formula obtained for the gradient of velocity L2 error is of order :
Lift technique gives one higher order for the velocity L2 error.
Thus, we see that we expect to get 3 and 2 as the convergence rates for velocity and gradient of the velocity, respectively.
Table 1. Errors for order of deconvolution N=0 and χ = 0.01
Table 2. Errors for order of deconvolution N=1 and χ = 0.01
Literature Cited
[1] N. A. Adams and S. Stolz, Deconvolution methods for subgrid-scale approximation in large eddy simulation, Modern Simulation Strategies for Turbulent Flow, R.T. Edwards, 2004.
[2] V. Ervin, W. J. Layton and Monika, Numerical Analysis of a higher Order Time Relaxation Model of Fluids , International Journal of Numerical Analysis and Modeling, 4: , 2006. h
|| ||
rate
|| ||
1/40
8.4572e-06
8.6365e-04
1/50
3.8429e-06
3.53
4.9923e-04
2.46
1/60
2.0549e-06
3.43
3.2444e-04
2.36
1/70
1.2262e-06
3.35
2.2786e-04
2.29
1/80
7.9165e-07
3.28
1.6901e-04
2.24
For further information h
|| ||
rate
|| ||
1/40
8.4568e-06
8.6365e-04
1/50
3.8427e-06
3.53
4.9923e-04
2.45
1/60
2.0547e-06
3.43
3.2444e-04
2.36
1/70
1.2261e-06
3.35
2.2786e-04
2.29
1/80
7.9156e-07
3.28
1.6901e-04
2.24
If you have any questions or would like more information, the authors may be reached at and