Hutton - Fundamentals of Finite Element Analysis (523155), страница 58
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Table 8.1 lists values of obtained by the finite element solution and the preceding analytical solution at several selected nodes in the model. Thecomputed magnitude of the fluid velocity at those points is also given. The nominal errorsin the finite element solution versus the analytical solution are about 4 percent for thevalue of the stream function and 6 percent for the velocity magnitude. While not shownhere, a refined element mesh consisting of 218 elements was used in a second solutionand the errors decreased to less than 1 percent for both the stream function value and thevelocity magnitude.Earlier in the chapter, the analogy between the heat conduction problem andthe stream function formulation is mentioned.
It may be of interest to the readerto note that the stream function solution presented in Example 8.1 is generatedusing a commercial software package and a two-dimensional heat transferelement. The particular software does not contain a fluid element of the typerequired for the problem. However, by setting the thermal conductivities to unityand specifying zero internal heat generation, the problem, mathematically, is thesame. That is, nodal temperatures become nodal values of the stream function.Similarly, spatial derivatives of temperature (flux values) become velocity components if the appropriate sign changes are taken into account.
The mathematicalsimilarity of the two problems is further illustrated by the finite element solutionof the previous example using the velocity potential function.EXAMPLE 8.2Obtain a finite element solution for the problem of Example 8.1 via the velocity potentialapproach, using, specifically, the heat conduction formulation modified as required.Hutton: Fundamentals ofFinite Element Analysis8. Applications in FluidMechanics8.4Text© The McGraw−HillCompanies, 2004The Velocity Potential Function in Two-Dimensional Flow■ SolutionFirst let us note the analogiesu=−∂∂T⇒ q x = −k x∂x∂xv=−∂T∂⇒ q y = −k y∂y∂yso that, if k x = k y = 1 , then the velocity potential is directly analogous to temperature andthe velocity components are analogous to the respective flux terms.
Hence, the boundaryconditions, in terms of thermal variables becomeqx = Uqy = 0qx = q y = 0on a-bon b-c and a-e-dT = constant = 0on c-d (the value is arbitrary)Figure 8.8 shows a coarse mesh finite element solution that plots the lines of constantvelocity potential (in the thermal solution, these lines are lines of constant temperature,419202141473752233234635403227 3930253644142458284315263134324141312291179102Figure 8.8 Lines of constant velocity potential for the finiteelement solution of Example 8.2.311Hutton: Fundamentals ofFinite Element Analysis3128. Applications in FluidMechanicsCHAPTER 8Text© The McGraw−HillCompanies, 2004Applications in Fluid MechanicsTable 8.2 Velocity Components at Selected Nodes in Example 8.2Nodeuv4192021540.42341.01942.30943.33943.6760.4800.5270.5940.5160.002or isotherms).
A direct comparison between this finite element solution and that describedfor the stream function approach is not possible, since the element meshes are different.However, we can assess accuracy of the velocity potential solution by examination of theresults in terms of the boundary conditions. For example, along the upper horizontalboundary, the y-velocity component must be zero, from which it follows that lines of constant must be perpendicular to the boundary. Visually, this condition appears to be reasonably well-satisfied in Figure 8.8. An examination of the actual data presents a slightlydifferent picture.
Table 8.2 lists the computed velocity components at each node along theupper surface. Clearly, the values of the y-velocity component v are not zero, so additional solutions using refined element meshes are in order.Observing that the stream function and velocity potential methods areamenable to solving the same types of problems, the question arises as to whichshould be selected in a given instance. In each approach, the stiffness matrix isthe same, whereas the nodal forces differ in formulation but require the samebasic information. Hence, there is no significant difference in the two procedures.
However, if one uses the stream function approach, the flow is readilyvisualized, since velocity is tangent to streamlines. It can also be shown [2] thatthe difference in value of two adjacent streamlines is equal to the flow rate (perunit depth) between those streamlines.8.4.1 Flow around Multiple BodiesFor an ideal (inviscid, incompressible) flow around multiple bodies, the streamfunction approach is rather straightforward to apply, especially in finite elementanalysis, if the appropriate boundary conditions can be determined.
To beginthe illustration, let us reconsider flow around a cylinder as in Example 8.1. Observing that Equation 8.11 governing the stream function is linear, the principleof superposition is applicable; that is, the sum of any two solutions to the equation is also a solution. In particular, we consider the stream function to be givenby(x , y) = 1 (x , y) + a 2 (x , y)(8.42)where a is a constant to be determined.
The boundary conditions at the horizontal surfaces ( S1 ) are satisfied by 1 , while the boundary conditions on the surfaceHutton: Fundamentals ofFinite Element Analysis8. Applications in FluidMechanics8.4Text© The McGraw−HillCompanies, 2004The Velocity Potential Function in Two-Dimensional Flowof the cylinder ( S2 ) are satisfied by 2 . The constant a must be determined so thatthe combination of the two stream functions satisfies a known condition at somepoint in the flow. Hence, the conditions on the two solutions (stream functions)are∂ 2 1∂ 2 1+=0∂x2∂ y2(everywhere in the domain)∂ 2 2∂ 2 2+=0∂x2∂ y2(everywhere in the domain)(8.43) 1 = U ybon S1(8.44)1 = 0on S2(8.45)2 = 0on S1(8.46)2 = 1on S2(8.47)Note that the value of 2 is (temporarily) set equal to unity on the surface of thecylinder.
The procedure is then to obtain two finite element solutions, one foreach stream function, and associated boundary conditions. Given the two solutions, the constant a can be determined and the complete solution known. Theconstant a, for example, is found by computing the velocity at a far upstreamposition (where the velocity is known) and calculating a to meet the knowncondition.In the case of uniform flow past a cylinder, the solutions give the trivial resultthat a = arbitrary constant, since we have only one surface in the flow, hence onearbitrary constant.
The situation is different if we have multiple bodies, however,as discussed next.Consider Figure 8.9, depicting two arbitrarily shaped bodies located in anideal fluid flow, which has a uniform velocity profile at a distance upstreamfrom the two obstacles. In this case, we consider three solutions to the governingUS1S2S3Figure 8.9 Two arbitrary bodies in a uniform stream.
Theboundary conditions must be specified on S1, S2, and S3 within aconstant.313Hutton: Fundamentals ofFinite Element Analysis3148. Applications in FluidMechanicsCHAPTER 8Text© The McGraw−HillCompanies, 2004Applications in Fluid Mechanicsequation, so that the stream function can be represented by [3](x , y) = 1 (x , y) + a 2 (x , y) + b 3 (x , y)(8.48)where a and b are constants to be determined.
Again, we know that each independent solution in Equation 8.48 must satisfy Equation 8.11 and, recalling thatthe stream function must take on constant value on an impenetrable surface, wecan express the boundary conditions on each solution as 1 = U ybon S11 = 0on S2 and S32 = 0on S1 and S32 = 1on S23 = 0on S1 and S23 = 1on S3(8.49)To obtain a solution for the flow problem depicted in Figure 8.9, we must1.
Obtain a solution for 1 satisfying the governing equation and the boundaryconditions stated for 1 .2. Obtain a solution for 2 satisfying the governing equation and the boundaryconditions stated for 2 .3. Obtain a solution for 3 satisfying the governing equation and the boundaryconditions stated for 3 .4. Combine the results at (in this case) two points, where the velocity orstream function is known in value, to determine the constants a and b inEquation 8.48. For this example, any two points on section a-b are appropriate, as we know the velocity is uniform in that section.As a practical note, this procedure is not generally included in finite elementsoftware packages. One must, in fact, obtain the three solutions and hand calculate the constants a and b, then adjust the boundary conditions (the constant values of the stream function) for entry into the next run of the software.
In this case,not only the computed results (stream function values, velocities) but the valuesof the computed constants a and b are considerations for convergence of thefinite element solutions. The procedure described may seem tedious, and it is toa certain extent, but the alternatives (other than finite element analysis) are muchmore cumbersome.8.5 INCOMPRESSIBLE VISCOUS FLOWThe idealized inviscid flows analyzed via the stream function or velocity potential function can reveal valuable information in many cases. Since no fluid istruly inviscid, the accuracy of these analyses decreases with increasing viscosityHutton: Fundamentals ofFinite Element Analysis8. Applications in FluidMechanicsText© The McGraw−HillCompanies, 20048.5 Incompressible Viscous Flowof a real fluid. To illustrate viscosity effects (and the arising complications) wenow examine application of the finite element method to a restricted class ofincompressible viscous flows.The assumptions and restrictions applicable to the following developmentsare1.2.3.4.The flow can be considered two dimensional.No heat transfer is involved.Density and viscosity are constant.The flow is steady with respect to time.Under these conditions, the famed Navier-Stokes equations [4, 5], representingconservation of momentum, can be reduced to [6]u∂u∂u∂ 2u∂ 2u∂p+ v− 2 − 2 += FBx∂x∂y∂x∂y∂x∂v∂ 2v∂p∂v∂ 2vu+ v− 2 − 2 += FBy∂x∂y∂x∂y∂y(8.50)whereu and v = x-, and y-velocity components, respectively = density of the fluidp = pressure = absolute fluid viscosityFBx , FBy = body force per unit volume in the x and y directions, respectivelyNote carefully that Equation 8.50 is nonlinear, owing to the presence of the convective inertia terms of the form u(∂ u/∂ x ) .
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