Hutton - Fundamentals of Finite Element Analysis (523155), страница 56
Текст из файла (страница 56)
To understand the significance of the latter point,reexamine Equation 7.23 and set = T , k x = k y = 1, Q = 0 , and h = 0 . Theresult is the Laplace equation governing the stream function.The stream function over the domain of interest is discretized into finiteelements having M nodes:M(x , y) =N i (x , y) i = [N ]{}(8.14)i=1Using the Galerkin method, the element residual equations are 2∂ ∂ 2N i (x , y)+dx dy = 0i = 1, M∂x2∂ y2(8.15)A(e)or[N ] TA(e)∂ 2∂ 2+2∂x∂ y2dx dy = 0(8.16)Application of the Green-Gauss theorem gives∂∂ [N ] T ∂ ∂[N ] Tn x dS −dx dy +n y dS[N ] T∂x∂x ∂x∂yS (e)A(e)−A(e)S (e)∂ [N ] T ∂ dx dy = 0∂y ∂y(8.17)where S represents the element boundary and (n x , n y ) are the components of theoutward unit vector normal to the boundary. Using Equations 8.10 and 8.14results in ∂ [N ] T ∂ [N ]∂ [N ] T ∂ [N ]+dx dy {} =[N ] T (un y − vn x ) dS∂x∂x∂y∂yA(e)S (e)(8.18)and this equation is of the formk (e) {} = f (e)(8.19)299Hutton: Fundamentals ofFinite Element Analysis3008.
Applications in FluidMechanicsCHAPTER 8Text© The McGraw−HillCompanies, 2004Applications in Fluid MechanicsThe M × M element stiffness matrix is (e) ∂ [N ] T ∂ [N ]∂ [N ] T ∂ [N ]k=+dx dy∂x∂x∂y∂y(8.20)A(e)and the nodal forces are represented by the M × 1 column matrix (e) f=[N ] T (un y − vn x ) dS(8.21)S (e)Since the nodal forces are obtained via integration along element boundaries andthe unit normals for adjacent elements are equal and opposite, the forces oninterelement boundaries cancel during the assembly process. Consequently, theforces defined by Equation 8.21 need be computed only for element boundariesthat lie on global boundaries.
This observation is in keeping with similar observations made previously in context of other problem types.8.3.2 Boundary ConditionsAs the governing equation for the stream function is a second-order, partial differential equation in two independent variables, four boundary conditions mustbe specified and satisfied to obtain the solution to a physical problem.
The manner in which the boundary conditions are applied to a finite element model isdiscussed in relation to Figure 8.4a. The figure depicts a flow field between twoparallel plates that form a smoothly converging channel. The plates are assumedsufficiently long in the z direction that the flow can be adequately modeled astwo-dimensional.
Owing to symmetry, we consider only the upper half of theflow field, as in Figure 8.4b. Section a-b is assumed to be far enough from theconvergent section that the fluid velocity has an x component only. Since we examine only steady flow, the velocity at a-b is U ab = constant. A similar argumentapplies at section c-d, far downstream, and we denote the x-velocity componentat that section as U cd = constant. How far upstream or downstream is enough tomake these assumptions? The answer is a question of solution convergence. Thedistances involved should increase until there is no discernible difference in theflow solution. As a rule of thumb, the distances should be 10–15 times the widthof the flow channel.As a result of the symmetry and irrotationality of the flow, there can be novelocity component in the y direction along the line y = 0 (i.e., the x axis).
Thevelocity along this line is tangent to the line at all values of x. Given these observations, the x axis is a streamline; hence, = 1 = constant along the axis.Similarly, along the surface of the upper plate, there is no velocity componentnormal to the plate (imprenetrability), so this too must be a streamline alongwhich = 2 = constant. The values of 1 and 2 are two of the requiredboundary conditions.
Recalling that the velocity components are defined asfirst partial derivatives of the stream function, the stream function must beknown only within a constant. For example, a stream function of the formHutton: Fundamentals ofFinite Element Analysis8. Applications in FluidMechanicsText© The McGraw−HillCompanies, 20048.3 The Stream Function in Two-Dimensional FlowybUabcaxd(a)2bUabc1aUcdd(b)bcad(c)Figure 8.4(a) Uniform flow into a converging channel. (b) Half-symmetrymodel showing known velocities and boundary values of the streamfunction.
(c) A relatively coarse finite element model of the flowdomain, using three-node triangular elements. This model includes65 degrees of freedom before applying boundary conditions.(x , y) = C + f (x , y) contributes no velocity terms associated with the con-stant C. Hence, one (constant) value of the stream function can be arbitrarilyspecified. In this case, we choose to set 1 = 0. To determine the value of 2 , wenote that, at section a-b (which we have arbitrarily chosen as x = 0 , the velocityisu=∂2 − 12= U ab = constant ==∂yyb − yayb(8.22)301Hutton: Fundamentals ofFinite Element Analysis3028.
Applications in FluidMechanicsCHAPTER 8Text© The McGraw−HillCompanies, 2004Applications in Fluid Mechanicsso 2 = yb U ab . At any point on a-b, we have = ( 2 /yb ) y = U ab y , so thevalue of the stream function at any finite element node located on a-b is known.Similarly, it can be shown that = ( 2 /yc ) y = U ab ( yb /yc ) y along c-d, so nodalvalues on that line are also known.
If these arguments are carefully considered,we see that the boundary conditions on at the “corners” of the domain are continuous and well-defined.Next we consider the force conditions across sections a-b and c-d. As noted,the y-velocity components along these sections are zero. In addition, the y components of the unit vectors normal to these sections are zero as well. Using theseobservations in conjunction with Equation 8.21, the nodal forces on any elementnodes located on these sections are zero. The occurrence of zero forces is equivalent to stating that the streamlines are normal to the boundaries.If we now utilize a mesh of triangular elements (for example), as in Figure 8.4c, and follow the general assembly procedure, we obtain a set of globalequations of the form[K ]{} = {F }(8.23)The forcing function on the right-hand side is zero at all interior nodes.
At theboundary nodes on sections a-b and c-d, we observe that the nodal forces are zeroalso. At all element nodes situated on the line y = 0 , the nodal values of thestream function are = 0 , while at all element nodes on the upper plate profilethe values are specified as = yb U ab . The = 0 conditions are analogous tothe specification of zero displacements in a structural problem. With such conditions, the unknowns are the forces exerted at those nodes.
Similarly, the specification of nonzero value of the stream function along the upper plate profileis analogous to a specified displacement. The unknown is the force required toenforce that displacement.The situation here is a bit complicated mathematically, as we have both zeroand nonzero specified values of the nodal variable. In the following, we assumethat the system equations have been assembled, and we rearrange the equationssuch that the column matrix of nodal values is { 0 } {} = { s }(8.24){ u }where { 0 } represents all nodes along the streamline for which = 0, { s } represents all nodes at which the value of is specified, and { u } corresponds to allnodes for which is unknown.
The corresponding global force matrix is {F0 } {F} = {Fs }(8.25){0}and we note that all nodes at which is unknown are internal nodes at which thenodal forces are known to be zero.Hutton: Fundamentals ofFinite Element Analysis8. Applications in FluidMechanicsText© The McGraw−HillCompanies, 20048.3 The Stream Function in Two-Dimensional FlowUsing the notation just defined, the system equations can be rewritten (bypartitioning the stiffness matrix) as [K 00 ] [K 0s ] [K 0u ] { 0 } {F0 } [K s0 ] [K ss ] [K su ] { s } = {Fs }(8.26) [K u0 ] [K us ] [K uu ]{ u }{0}Since 0 = 0, the first set of partitioned equations become[K 0s ]{ s } + [K 0u ]{ u } = {F0 }(8.27)and the values of F0 can be obtained only after solving for { u } using the remaining equations. Hence, Equation 8.27 is analogous to the reaction force equations in structural problems and can be eliminated from the system temporarily.The remaining equations are [K ss ] [K su ]{ s }{Fs }=(8.28)[K us ] [K uu ]{ u }{0}and it must be noted that, even though the stiffness matrix is symmetric, [K su ]and [K us ] are not the same.
The first partition of Equation 8.28 is also a set of“reaction” equations given by[K ss ] { s } + [K su ] { u } = {Fs }(8.29)and these are used to solve for { Fs } but, again, after { u } is determined. The second partition of Equation 8.28 is[K us ] { s } + [K uu ] { u } = {0}(8.30)and these equations have the formal solution{ u } = −[K uu ]−1 [K us ] { s }(8.31)since the values in { s } are known constants.
Given the solution represented byEquation 8.31, the “reactions” in Equations 8.27 and 8.28 can be computeddirectly.As the velocity components are of major importance in a fluid flow, we mustnext utilize the solution for the nodal values of the stream function to computethe velocity components. This computation is easily accomplished given Equation 8.14, in which the stream function is discretized in terms of the nodal values.Once we complete the already described solution procedure for the values of thestream function at the nodes, the velocity components at any point in a specifiedfinite element areu(x , y) =M∂∂ Ni∂ [N ] T{}=i =∂y∂y∂yi=1M∂∂ Ni∂ [N ] T{}v(x , y) = −=−i = −∂x∂x∂xi=1(8.32)303Hutton: Fundamentals ofFinite Element Analysis3048.
Applications in FluidMechanicsCHAPTER 8Text© The McGraw−HillCompanies, 2004Applications in Fluid MechanicsNote that if, for example, a three-node triangular element is used, the velocitycomponents as defined in Equation 8.32 have constant values everywhere in theelement and are discontinuous across element boundaries. Therefore, a largenumber of small elements are required to obtain solution accuracy. Applicationof the stream function to a numerical example is delayed until we discuss analternate approach, the velocity potential function, in the next section.8.4 THE VELOCITY POTENTIAL FUNCTIONIN TWO-DIMENSIONAL FLOWAnother approach to solving two-dimensional incompressible, inviscid flowproblems is embodied in the velocity potential function. In this method, wehypothesize the existence of a potential function (x , y) such thatu(x , y) = −∂∂xv(x , y) = −∂∂y(8.33)and we note that the velocity components defined by Equation 8.33 automatically satisfy the irrotationality condition.
Substitution of the velocity definitionsinto the continuity equation for two-dimensional flow yields∂ 2∂u∂v∂ 2+=+=02∂x∂y∂x∂ y2(8.34)and, again, we obtain Laplace’s equation as the governing equation for 2-D flowdescribed by a potential function.We examine the potential formulation in terms of the previous example of aconverging flow between two parallel plates. Referring again to Figure 8.4a, wenow observe that, along the lines on which the potential function is constant, wecan writed =∂∂dx +dy = −(u dx + v dy) = 0∂x∂y(8.35)Observing that the quantity u dx + v dy is the magnitude of the scalar product ofthe velocity vector and the tangent to the line of constant potential, we concludethat the velocity vector at any point on a line of constant potential is perpendicular to the line. Hence, the streamlines and lines of constant velocity potential(equipotential lines) form an orthogonal “net” (known as the flow net) as depicted in Figure 8.5.The finite element formulation of an incompressible, inviscid, irrotationalflow in terms of velocity potential is quite similar to that of the stream functionapproach, since the governing equation is Laplace’s equation in both cases.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
