Hutton - Fundamentals of Finite Element Analysis (523155), страница 60
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Associated with each specified velocityis an unknown “reaction” force represented by the shear stress-related forces in Equations 8.56, and these forces can be computed using the constraint equations after theglobal solution is obtained. This is the case for all equations associated with elementnodes on segments S1, S2, and S3. On S4, the situation is a little different and additionalcomment is warranted. As the velocity components in the x direction along S4 are notspecified, a question arises as to the disposition of the shear-related forces in the x direction. These forces are given byfx = NiS (e)∂u∂unx +ny∂x∂ydSas embodied in Equation 8.57. On the boundary in question, the unit outward normal vector is defined by (n x , n y ) = (0, −1) , so the first term in this integral is zero. In view of thesymmetry conditions about a-c, we also have ∂ u/∂ y = 0 , so the shear forces in the xdirection along S4 are also zero.
With this observation and the boundary conditions, theglobal matrix equations become a tractable system of algebraic equations that can besolved for the unknown values of the nodal variables.8.5.2 Viscous Flow with InertiaHaving discussed slow flows, in which the inertia terms were negligible, we nowconsider the more general, nonlinear case. All the developments of the previous321Hutton: Fundamentals ofFinite Element Analysis3228.
Applications in FluidMechanicsCHAPTER 8Text© The McGraw−HillCompanies, 2004Applications in Fluid Mechanicssection on Stokes flow are applicable here; we now add the nonlinear terms arising from the convective inertia terms. From the first of Equation 8.50, we add aterm of the form ∂u∂u∂ [N ]∂ [N ]+vdA ⇒ {u} + [N ]{v}{u} d A u[N ]{u}∂x∂y∂x∂yA(e)A(e)(8.68)and from the second equation of 8.50, ∂v∂v∂ [N ]∂ [N ] u+vdA ⇒ [N ]{u}{v} + [N ]{v}{v} d A∂x∂y∂x∂yA(e)A(e)(8.69)As expressed, Equation 8.68 is not conformable to matrix multiplication, as inbeing able to write the expression in the form [k]{u} , and this is a direct result ofthe nonlinearity of the equations.
While a complete treatment of the nonlinearequations governing viscous fluid flow is well beyond the scope of this text, wediscuss an iterative approximation for the problem.Let us assume that for a particular two-dimensional geometry, we havesolved the Stokes (creeping) flow problem and have all the nodal velocities of theStokes flow finite element model available.
For each element in the finite element model, we denote the Stokes flow solution for the average velocity components (evaluated at the centroid of each element) as ( ū, v̄) ; then, we express theapproximation for the inertia terms (as exemplified by Equation 8.69) asA(e)∂u∂u u+v∂x∂ydA =A(e)∂ [N ]∂ [N ] ū+ v̄∂x∂yd A{u} = [k uv ]{u}(8.70)Similarly, we find the y-momentum equation contribution to beA(e)∂v∂v u+v∂x∂ydA =A(e)∂ [N ]∂ [N ] ū+ v̄∂x∂yd A{v} = [k vu ]{v}(8.71)Equations 8.70 and 8.71 refer to an individual element. The assembly proceduresare the same as discussed before; now we add additional terms to the stiffnessmatrix as a result of inertia. These terms are readily identifiable in Equations 8.70and 8.71.
In the viscous inertia flow, the solution requires iteration to achieve satisfactory results. The use of the Stokes flow velocities and pressures representonly the first iteration (approximation). At each iteration, the newly computedvelocity components are used for the next iteration.Hutton: Fundamentals ofFinite Element Analysis8. Applications in FluidMechanicsText© The McGraw−HillCompanies, 2004ReferencesFor both creeping flow and flow with inertia, the governing equations can alsobe developed in terms of a stream function [3]. However, the resulting (single)governing equation in each case is found to be fourth order. Consequently, elements exhibiting continuity greater than C 0 are required.8.6 SUMMARYApplication of the finite element method to fluid flow problems is, in one sense,quite straightforward and, in another sense, very complex. In the idealized casesof inviscid flow, the finite element problem is easily formulated in terms of a single variable.
Such problems are neither routine nor realistic, as no fluid is trulywithout viscosity. As shown, introduction of the very real property of fluid viscosity and the historically known, nonlinear governing equations of fluid flowmake the finite element method for fluid mechanics analysis difficult and cumbersome, to say the least.The literature of fluid mechanics is rife with research results on the application of finite element methods to fluid mechanics problems. The literature is sovoluminous, in fact, that we do not cite references, but the reader will find thatmany finite element software packages include fluid elements of various types.These include “pipe elements,” “acoustic fluid elements,” and “combinationelements.” The reader is warned to be aware of the restrictions and assumptionsunderlying the “various sorts” of fluid elements available in a given softwarepackage and use care in application.REFERENCES1. Halliday, D., R.
Resnick, and J. Walker. Fundamentals of Physics, 6th ed.New York: John Wiley and Sons, 1997.2. Crowe, C. T., J. A. Roberson, and D. F. Elger. Engineering Fluid Mechanics,7th ed. New York: John Wiley and Sons, 1998.3. Huebner, K. H., and E. A. Thornton. The Finite Element Method for Engineers.New York: John Wiley and Sons, 1982.4. Navier, M. “Memoire sur les Lois dur Mouvement des Fluides.” Mem.
De l’Acadd. Sci. (1827).5. Stokes, G. G. “On the Theories of Internal Friction of Fluids in Motion.”Transactions of the Cambridge Philosophical Society (1845).6. Schlichting, H. Boundary Layer Theory, 7th ed. New York: McGraw-Hill, 1979.7. Baker, A. J. Finite Element Computational Fluid Mechanics. New York: McGrawHill, 1983.8. Stasa, F. L. Applied Finite Element Analysis for Engineers. New York: Holt,Rinehart and Winston, 1985.323Hutton: Fundamentals ofFinite Element Analysis3248. Applications in FluidMechanicsCHAPTER 8Text© The McGraw−HillCompanies, 2004Applications in Fluid MechanicsPROBLEMS8.18.28.38.48.58.68.78.8Per the standard definition of viscosity described in Section 8.1, how would youdescribe the property of viscosity, physically, in terms of an everyday example(do not use water and molasses—I already used that example)?How would you design an experiment to determine the relative viscositybetween two fluids? What fluids might you use in this test?Look into a fluid mechanics text or reference book.
What is the definition of aNewtonian fluid?Equation 8.5 is a rather complicated partial differential equation, what doesit really mean? Explain how that equation takes the very simple form ofEquation 8.6.If you visually examine a fluid flow, could you determine whether it wasrotational or irrotational? Why? Why not?Why do we use the Green-Gauss theorem in going from Equation 8.16 toEquation 8.17? Refer to Chapter 5.Recalling that Equation 8.21 is based on unit depth in a two-dimensional flow,what do the nodal forces represent physically?Given the three-node triangular element shown in Figure P8.8, compute thenodal forces corresponding to the flow conditions shown, assuming unit depthinto the plane.3U(0, 1)1 (0, 0)2(1, 0)Figure P8.88.98.10Per Equation 8.32, how do the fluid velocity components vary withina.
A linear, three-node triangular element.b. A four-node rectangular element.c. A six-node triangular element.d. An eight-node rectangular element.e. Given questions a–d, how would you decide which element to use in a finiteelement analysis?We show, in this chapter, that both stream function and velocity potentialmethods are governed by Laplace’s equation.
Many other physical problemsare governed by this equation. Consult mathematical references and findother applications of Laplace’s equation. While you are at it (and learningHutton: Fundamentals ofFinite Element Analysis8. Applications in FluidMechanicsText© The McGraw−HillCompanies, 2004Problems8.11the history of our profession is part of becoming an engineer), find outabout Laplace.Consider the uniform (ideal) flow shown in Figure P8.11. Use the fourtriangular elements shown to compute the stream function and derive thevelocity components.
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