Computational Fluid Dynamics-vol1

8/7/2019 Computational Fluid Dynamics-vol1

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Fourth Edition

COMPUTATIONAL FLUID DYNAMICSVOLUME I

KLAUS A. HOFFMANNSTEVE T. CHIANG

onTO' K"OTOPH ANl';S1M. 1'.. T. U. L!J3RARY, ---------

A Publication of Engineering Education System,Wichita, Kansas, 6"/208-1078, USA

www.EESbooks.com


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6

Figure 1-1. Zone of influence (horizontal shading) and zone ofdependence (vertical shading) of point A.

1.4 Elliptic Equations

Chapter 1

A partial differential equation is elliptic in a region if (B 2 - 4AC) < 0 atall points of the region. An elliptic PDE has no real characteristic curves. Adisturbance is propagated instantly in all directions within the region. Examples ofelliptic equations are Laplace's equation

and Poisson's equation(1-9)

(1-10)

(1-11)

The domain of solution for an elliptic PDE is a closed region, R, shown in Figure1-2. On the closed boundary of R, either the value of the dependent variable, itsnormal gradient, or a linear combination of the two is prescribed. Providing theboundary conditions uniquely yields the solution within the domain.

1.5 Parabolic EquationsA partial differential equation is classified as parabolic if (B 2 - 4AC) = 0 at all

points of the region. The solution domain for a parabolic PDE is an open region, asshown in Figure 1-3. For a parabolic partial differential equation there exists onecharacteristic line. Unsteady heat conduction in one dimension

aT 82T8t = a 8x2


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206 Chapter 6One further comment. Equation (6-28) is equivalent to the coupled first-order

wave equations given byau av- =a -at axav au-=a -at ax

(6-31a)

(6-31b)Therefore a solution of the original model equation (6-28) may be obtained by

solving the first-order equations (6-31a) and (6-31b).In conclusion, when one broadly compares the implicit and explicit methods just

explored, it is clear that, for linear hyperbolic equations, the explicit formulationsprovide better solutions than implicit methods. The advantages of implicit methods(which are usually unconditionally stable) are lost, since large step sizes producepoor results.

6.6 Nonlinear ProblemThe majority of partial differential equations in fluid mec:llanics and heat trans

fer are nonlinear. The simple linear hyperbolic equation just investigated shouldprovide some foundation to approach the nonlinear hyperbolic equations. A classical nonlinear first-order hyperbolic equation is the inviscid Burgers equation, whichwill be used as a model equation to investigate various solution procedures.In this section, the numerical techniques presented earlier for the linear problemwill be applied to the nonlinear model equation. The inviscid Burgers equation is

au au-=-u-at axwhich, in a conservative form, may be expressed as

a;: = - :x ( ~ or

(6-32)

auat

aEax (6-33)

where E = u2 /2. Equation (6-32) can be interpreted as the propagation of a wavewith each point having a different velocity and eventually forming a discontinuityin the domain. This is similar to the formation of shock waves by a series of weakcompression waves.


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362 Chapter 9

9.2 Transformation of the Governing PartialDifferential Equations

The equations of fluid motion include the continuity, momentum and energyequations. For a single phase continuum flow, the transformation of this system ofequations will be presented in Chapter 11. In this section, a simple 2-D problem isproposed to familiarize the reader with the processes involved in the transformationof a PDE and the complexity of the resultant equation. It should be mentioned thatthe form and type of the transformed equation remains the same as the originalPDEj Le., if the original equation is parabolic, then the transformed equation isalso parabolic. A mathematical proof is given in Reference [l}-I]. Now, define thefollowing relations between the physical and computational spaces:

- ~ ( x , y )." - .,,(x, y)

The chain rule for partial differentiation yields the following expression:a a a a." a-=--+--ax ax ae ax a."

(9-1)(9-2)

(9-3)The partial derivatives will be denoted using the subscripts notation, Le., ! = ezoHence, a a a

ax = ez ae + "'z a."and similarly, a a aay = el/ ae + "hi a."Now consider a model PDE, such as

au au-+a-=Qax ay

(9-4)

(9-5)

(9-6)This equation may be transformed from physical space to computational space usingEquations (9-4) and (9-5). As a result,

which may be rearranged as(9-7)

\-


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Computational Fluid Dynamics-vol1