Hutton - Fundamentals of Finite Element Analysis (523155), страница 59
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Rather than treat the nonlinearterms directly at this point, we first consider the following special case.8.5.1 Stokes FlowFor fluid flow in which the velocities are very small, the inertia terms (i.e., thepreceding nonlinear terms) can be shown to be negligible in comparison to theviscous effects. Such flow, known as Stokes flow (or creeping flow), is commonlyencountered in the processing of high-viscosity fluids, such as molten polymers.Neglecting the inertia terms, the momentum equations become−∂ 2u∂ 2u∂p−+= FBx22∂x∂y∂x∂ 2v∂ 2v∂p− 2 − 2 += FBy∂x∂y∂y(8.51)315Hutton: Fundamentals ofFinite Element Analysis3168.
Applications in FluidMechanicsCHAPTER 8Text© The McGraw−HillCompanies, 2004Applications in Fluid MechanicsEquation 8.51 and the continuity condition, Equation 8.8, form a system ofthree equations in the three unknowns u(x, y), v(x, y), and p(x, y). Hence, a finiteelement formulation includes three nodal variables, and these are discretized asu(x , y) =MN i (x , y)u i = [N ] T {u}i=1v(x , y) =MN i (x , y)vi = [N ] T {v}(8.52)i=1p(x , y) =MN i (x , y) pi = [N ] T { p}i=1Application of Galerkin’s method to a two-dimensional finite element (assumedto have uniform unit thickness in the z direction) yields the residual equationsA(e)∂ 2u∂ 2u∂pN i − 2 − 2 +− FBx d A = 0∂x∂y∂xNiA(e)∂ 2v∂ 2v∂p− 2 − 2 +− FBy d A = 0∂x∂y∂y∂v∂uNi+dA = 0∂x∂yi = 1, M(8.53)A(e)As the procedures required to obtain the various element matrices are covered indetail in previous developments, we do not examine Equation 8.53 in its entirety.Instead, only a few representative terms are developed and the remaining resultsstated by inference.First, consider the viscous terms containing second spatial derivatives ofvelocity components such as− NiA(e)∂ 2u∂ 2u+∂x2∂ y2dAi = 1, M(8.54)which can be expressed as−A(e)∂∂x∂uNi∂x ∂∂u∂ Ni ∂ u∂ Ni ∂ u+NidA ++dA∂y∂y∂x ∂x∂y ∂yA(e)i = 1, M(8.55)Hutton: Fundamentals ofFinite Element Analysis8.
Applications in FluidMechanicsText© The McGraw−HillCompanies, 20048.5 Incompressible Viscous FlowApplication of the Green-Gauss theorem to the first integral in expression (8.55)yields ∂u∂u∂∂∂u∂uNi+Nid A = − Ninx +n y dS−∂x∂x∂y∂y∂x∂yA(e)S (e)i = 1, M(8.56)where S(e) is the element boundary and (nx, ny) are the components of the unitoutward normal vector to the boundary. Hence, the integral in expression (8.54)becomes 2∂ 2u∂u∂ u∂u− Ni+nx +n y dSd A = − Ni∂x2∂ y2∂x∂yA(e)S (e)+A(e)∂ Ni ∂ u∂ Ni ∂ u+∂x ∂x∂y ∂ydA(8.57)Note that the first term on the right-hand side of Equation 8.57 represents a nodalboundary force term for the element.
Such terms arise from shearing stress. Aswe observed many times, these terms cancel on interelement boundaries andmust be considered only on the global boundaries of a finite element model.Hence, these terms are considered only in the assembly step. The second integralin Equation 8.57 is a portion of the “stiffness” matrix for the fluid problem, andas this term is related to the x velocity and the viscosity, we denote this portionof the matrix [k u ] . Recalling that Equation 8.57 represents M equations, theintegral is converted to matrix form using the first of Equation 8.52 to obtain ∂ [N ] T ∂ [N ]∂ [N ] T ∂ [N ]+d A {u} = [k u ]{u}(8.58)∂x∂x∂y∂yA(e)Using the same approach with the second of Equation 8.53, the results aresimilar.
We obtain the analogous result 2∂ v∂ 2v∂v∂v− Ni+ 2 d A = − Ninx +n y dS∂x2∂y∂x∂yA(e)S (e)+A(e)∂ Ni ∂ v∂ Ni ∂ v+∂x ∂x∂y ∂ydAProceeding as before, we can write the area integrals on the right as ∂ [N ] T ∂ [N ]∂ [N ] T ∂ [N ]+d A {v} = [k v ]{v}∂x∂x∂y∂yA(e)(8.59)(8.60)317Hutton: Fundamentals ofFinite Element Analysis3188. Applications in FluidMechanicsCHAPTER 8Text© The McGraw−HillCompanies, 2004Applications in Fluid MechanicsConsidering next the pressure terms and converting to matrix notation, thefirst of Equation 8.53 leads to∂ [N ][N ] Td A { p} = [k px ]{ p}(8.61)∂xA(e)and similarly the second momentum equation contains∂ [N ][N ] Td A { p} = [k py ]{ p}∂y(8.62)A(e)The nodal force components corresponding to the body forces are readilyshown to be given by{ f Bx } =[N ] T FBx d AA(e){ f By } =(8.63)[N ] T FBy d AA(e)Combining the notation developed in Equations 8.58–8.63, the momentum equations for the finite element are[k u ]{u} + [k px ]{ p} = { f Bx } + { f x }[k v ]{v} + [k py ]{ p} = { f By } + { f y }(8.64)where, for completeness, the nodal forces corresponding to the integrals overelement boundaries S(e) in Equations 8.57 and 8.59 have been included.Finally, the continuity equation is expressed in terms of the nodal velocitiesin matrix form as∂ [N ]T ∂ [N ][N ]d A{u} +[N ] Td A{v} = [k u ]{u} + [k v ]{v} = 0 (8.65)∂x∂yA(e)A(e)where[k u ] = [k px ] =[N ] TA(e)[k v ] = [k py ] =∂ [N ]dA∂x∂ [N ][N ]dA∂y(8.66)TA(e)As formulated here, Equations 8.64 and 8.65 are a system of 3M algebraic equations governing the 3M unknown nodal values {u}, {v}, { p} and can be expressedHutton: Fundamentals ofFinite Element Analysis8.
Applications in FluidMechanicsText© The McGraw−HillCompanies, 20048.5 Incompressible Viscous Flowformally as the system [ku ] [0] [k px ] {u} { f Bx } (e) (e) (e) ␦= f [0] [kv ] [k py ] {v} = { f By } ⇒ k {0}{ p}[ku ] [kv ][0]319(8.67)where [k (e) ] represents the complete element stiffness matrix.
Note that the element stiffness matrix is composed of nine M × M submatrices, and although theindividual submatrices are symmetric, the stiffness matrix is not symmetric.The development leading to Equation 8.67 is based on evaluation of boththe velocity components and pressure at the same number of nodes. This isnot necessarily the case for a fluid element. Computational research [7] showsthat better accuracy is obtained if the velocity components are evaluated ata larger number of nodes than pressures. In other words, the velocity components are discretized using higher-order interpolation functions than the pressure variable.
For example, a six-node quadratic triangular element could beused for velocities, while the pressure variable is interpolated only at the corner nodes, using linear interpolation functions. In such a case, Equation 8.66does not hold.The arrangement of the equations and associated definition of the elementstiffness matrix in Equation 8.67 is based on ordering the nodal variables as{␦} T = [u 1u2u3v1v2v3p1p2p3 ](using a three-node element, for example). Such ordering is well-suited to illustrate development of the element equations.
However, if the global equations fora multielement model are assembled and the global nodal variables are similarlyordered, that is,{} T = [U 1U 2 · · · V1V2 · · · P1P2 · · · PN ]the computational requirements are prohibitively inefficient, because the globalstiffness has a large bandwidth. On the other hand, if the nodal variables areordered as{} T = [U 1V1P1U2V2P2 · · · U NVNPN ]computational efficiency is greatly improved, as the matrix bandwidth is significantly reduced. For a more detailed discussion of banded matrices and associatedcomputational techniques, see [8].EXAMPLE 8.3Consider the flow between the plates of Figure 8.4 to be a viscous, creeping flow anddetermine the boundary conditions for a finite element model.
Assume that the flow isfully developed at sections a-b and c-d and the constant volume flow rate per unit thickness is Q.Hutton: Fundamentals ofFinite Element Analysis3208. Applications in FluidMechanicsCHAPTER 8Text© The McGraw−HillCompanies, 2004Applications in Fluid Mechanicsybcaxdc⬘b⬘(a)S2S1S3S4(b)Figure 8.10(a) Velocity of fully developed flow. (b) Boundary conditions.■ SolutionFor fully developed flow, the velocity profiles at a-b and c-d are parabolic, as shown inFigure 8.10a. Denoting the maximum velocities at these sections as U ab and U cd , we haveu(x a , y) = U ab 1 −v(x a , y) = 0u(x c , y) = U cd 1 −y2yb2y2yb2v(x c , y) = 0The volume flow rate is obtained by integrating the velocity profiles asybQ=2ydu(x a , y) dy = 20u(x c , y) dy0Substituting the velocity expressions and integrating yieldsU ab =3Q4ybU cd =3Q4ydand thus the velocity components at all element nodes on a-b and c-d are known.Hutton: Fundamentals ofFinite Element Analysis8.
Applications in FluidMechanicsText© The McGraw−HillCompanies, 20048.5 Incompressible Viscous FlowNext consider the contact between fluid and plate along b-c. As in cases of inviscidflow discussed earlier, we invoke the condition of impenetrability to observe that velocity components normal to this boundary are zero.
In addition, since the flow is viscous,we invoke the no-slip condition, which requires that tangential velocity components alsobe zero at the fluid-solid interface. Hence, for all element nodes on b-c, both velocitycomponents ui and vi are zero.The final required boundary conditions are obtained by observing the condition ofsymmetry along a-d, where v = v(x , 0) = 0 . The boundary conditions are summarizednext in reference to Figure 8.10b:U I = U ab 1 −y I2yb2y2U I = U cd 1 − I2ydU I = VI = 0VI = 0VI = 0on S1 (a-b)VI = 0on S2 (b-c)on S3 (c-d)on S4 (a-d)where I is an element node on one of the global boundary segments.The system equations corresponding to each of the specified nodal velocities justsummarized become constraint equations and are eliminated via the usual proceduresprior to solving for the unknown nodal variables.
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