COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Development of an Educational Flow Simulation System
Tameo Nakanishi *, Hironori Shibata, Mutsumi Sato
Department of Mechanical Systems Engineering, Yamagata University, Yonezawa, 9928510 Japan
Email: [email protected]
Abstract we developed an interactive two-dimensional flow simulation system. The system solves incompressible
laminar and turbulent flows around multiple bodies of complex geometry. The system is based on an overset grid
method, a semi-Lagrangian scheme, and a newly developed one-equation turbulence model. The system integrates all
necessary features of flow simulation, and is easy-to-use. Using the flow system is similar to that of using a drawing
software. The control points of an arbitrary body shape are sequentially inputted by mouse. The grids are automatically
generated. The flow is visualized by velocity vectors, pressure and velocity contours, and streamlines. The system
enables qualitative real-time simulation by use of large time step length and quantitative simulation by use of adequate
time step length of a specified flow problem.
Key words: computational fluid dynamics, educational software, incompressible flow, overset grid method
INTRODUCTION
In late years computational fluid dynamics (CFD) progresses drastically and becomes an important tool of engineering
design. Various flow simulation systems have been developed and gained big success. Effective use of these systems,
however, requires background of fluid mechanics and numerical analysis. These systems are still expensive to be used
for college education. To meet the demand of engineering education, we developed an interactive two-dimensional
flow simulation system. The system solves incompressible laminar and turbulent flows around multiple bodies of
complex geometry. The core technology of the system is based on an overset grid method [1], a semi-Lagrangian
scheme of third-order accuracy [2,3], and a one-equation turbulence model [4] newly developed by the first author.
The system integrates all necessary functions of flow simulation, and is robust enough that anyone can easily handle it.
The system enables qualitative real-time simulation by use of large time step length and quantitative simulation by use
of adequate time step length of a specified flow problem.
Using the flow system is similar to that of using the Microsoft PowerPoint to draw a picture. The control points of an
arbitrary body shape are sequentially inputted by mouse. The system allows addition, deletion and moving of control
points. Dragging, deforming, rotating and copying of bodies are supported. As sample shapes, a circle, a square,
NACA 4-digit airfoils, a car shape are prepared. These shapes can be modified to make new shapes. The main-grid
covering the whole computational region and the sub-grid around each body are automatically generated. The flow is
visualized by velocity vectors, pressure and velocity contours, and streamlines. The pressure and shear stress plots of
body surface, the time-plot of aerodynamic coefficients are supported.
GOVERNING EQUATIONS AND NUMERICAL METHOD
The continuity equation and the incompressible Navier-Stokes equations are employed as governing equations.
where u is the velocity vector, p is the pressure, and Re is the Reynolds number.
When a turbulent flow is considered, the alternative Reynolds averaged Navier-Stokes equation as well as an original
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one-equation turbulence model[4] for the eddy viscosity (vt) are applied.
Here we only discuss the laminar flow case for the sake of simplicity. The governing equations (1) and (2) are
transformed into and solved in the generalized coordinates. The time integration of the governing equations is
conducted through four steps[2,3].
where the superscript n denotes time step. The remaining superscripts express the intermediate time.
Eqs. (7), (9), and (10) are solved by the SOR method. The advection equation (8) is calculated by a third-order
semi-Lagrangian scheme [2, 3] so that an arbitrarily large time step length can be applied. The idea of a
semi-Lagrangian method for an advection equation is described as follows. The solution of the unknown function f of
an advection equation (12) at the time instant t +Δt , the spatial location r≡(x, y) equals to the value of f at the time
instant t , the spatial location rUDP (see Fig. 1).
Figure 1: Illustration of semi-Lagrangian method
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The point of rUDP is called the Upstream Departure Point (UDP) and the path from rUDP to r the trajectory. Interpolating
the unknown from the grid cell currently enclosing the UDP, the explicit scheme independent of the CFL condition can
be constructed. In the present paper, the UDP is calculated in the generalized coordinates by the four-stage fourth order
Runge-Kutta method. The velocity vector at the UDP is interpolated by the cubic interpolation function of Hermite
type.
The overset grid method [1] (Fig. 2) is applied to treat the flow around multiple bodies. The main-grid covering the
whole computational region and the sub-grid around each body are employed. Communication of data between
different grids is made by bi-linear interpolation at each time step and inner iteration loops. The sub-grid around each
body is automatically created by an algebraic grid generation method (Fig. 3), which first expends the contour of the
body in the normal direction and then conducts relaxation of curvature of the new contour.
Figure 2: schematic of overset grid method
Figure 3: schematic of algebraic grid generation method
DESCRIPTION OF SYSTEM AND NUMERICAL EXAMPLES
Figure 4: operation flow chart of the system
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Fig. 5. Double clicking the mouse completes a body configuration. Points can be added (left key of mouse), deleted
(right key of mouse), and dragged to any location. Sample shape may be created by choosing sample shape menus such
as NACA 4-digit airfoil menu (Fig. 6). The selected sample shape may be hereafter modified by adding/
deleting/dragging of control points.
Figure 5: mouse input of arbitrary body shape
Figure 6: NACA 4-digit airfoil shape input menu
Selecting
menu, there appears the grid generation dialog window of Fig. 7. The grid parameters such as
number of grid points in each direction, the smallest grid space, and the distribution of grid points can be inputted.
Default values are also provided so that grid can be created by just pressing the
button. The thickness of the
sub-grid may be hereafter adjusted by clicking the control point (Fig. 8).
Figure 7: grid generation dialog window
Figure 9: sub-grid guideline
Figure 8: thickness adjustment of sub-grid
Figure 10: three bodies placed in computational region
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Selecting
menu, the body/ sub-grid can be placed at any location in the computational region. Multiple
bodies may be generated and located. As shown in Fig. 9, Copy/Rotate/Delete/Deform of an existing body/sub-grid
can be done by clicking each corner point of the sub-grid guideline. Fig. 10 shows an example where three bodies are
placed in computational region.
As shown in Fig. 11, flow and computational conditions such as Reynolds number, time step length, total number of
and the
time integration steps, boundary conditions and etc. can be inputted by choosing
following sub-menus: flow condition input dialog and boundary condition input dialog. Default values are also
provided so that these operations may be skipped.
Figure 11: (a) flow condition input dialog (left), and (b) boundary condition input dialog (right)
The result visualization window is shown in Fig. 12. By pressing Execute/Control button and the sequential start
computation button, flow computation starts and computational results are automatically displayed in pressure
contours and surface pressure plots. Clicking the object switches the object of surface pressure plots. The flow
conditions, maximum and minimum values of the velocity and pressure are displayed in data. Clicking arbitrary point
in the computational region, the coordinates, velocity and pressure information are displayed. Qualitative real-time
computation/displaying of the flow field is possible by using large time step length. Plots of aerodynamic coefficients
( Cl,Cd ) against time are also supported by choosing corresponding sub-menu under the Result visualization menu.
surface pressure
Figure 12: result visualization window (displayed as pressure contours)
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The result visualization tools of the present system are shown in Fig. 13. Velocity vectors, velocity and pressure
contours, and streamlines can be displayed. Fig. 14 shows applications to problems of multiple body and complex
body geometry. The accuracy of the present system has been verified by the flow around the circular cylinder at
Re=100 and 1000. By use of a time step length of _t = 0.005 for each case, numerical results in terms of the drag
coefficient as well as the Strouhal number agree well with the experimental data. Demonstration of the system will be
presented at the conference.
(a) grids
(b) velocity vectors
(c) velocity contours
(d) streamlines
(e) pressure contours
(f) pressure contours in flood pattern
Figure 13: result visualization tools
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Figure 14: applications to problems of multiple body and complex body geometry
CONCLUSIONS
An interactive two-dimensional flow simulation system for educational use has been developed. The system solves
incompressible laminar and turbulent flows around multiple bodies of complex geometry. The system integrates all
necessary functions such as body shape input by mouse clicking, automatic grid generation, flow calculation, result
display, and so on. The system is powerful, flexible, and easy-to-use. The ability of the system were demonstrated by
examples.
REFERENCES
1. Jia W, Nakamura Y. Incompressible flow solver of arbitrarily moving bodies with rigid surface. JSME
International Journal, Series B, 1996; 39: 315-325.
2. Jia W. An accurate semi-Lagrangian scheme designed for incompressible Navier-Stokes equations written in
generalized coordinates. Trans. Jan. Soc. Aero. Space Sci., 1998; 41: 105-117.
3. Nakanishi T, Abe M. Computing high Reynolds number flows by a semi-Lagrangian scheme. Trans. of Japan
Society for Aeronautical and Space Sciences, 2001; 44(143): 13-23.
4. Nakanishi T. A wall-distance free one-equation turbulence model. (in preparation).
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