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HW15sols - Problem 5.1[Difficultyz 1 5.1 Which of the...

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Unformatted text preview: Problem 5.1 [Difficultyz 1] 5.1 Which of the following sets of equations represent pos- sible two-dimensional incompressible flow cases? (a) u, = 2:2 +192 —*r1}t; v = 3:3 + - 4}») (b’) u=2xy—x2y; v=2ry—yz+x2 (c) u = f: +2y; v = x1? —yt (d) u a (2x + dfixz; v a —3(x + fly: Given: The list of velocity fields provided above F ind: Which of these fields possibly represent two-dimensional, incompressible flow Solution: We will check these flow fields against the continuity equation Governing 6 Equations: —— ‘ 8x (1) Incompressible' flow (p is constant) (2) Two dimensional flow (velocity is not a fimction of z) (pu) + — + — + a—’0 = 0 (Continuity equation) Assumptions: 1 Based on the two assumptions listed above, the continuity equation reduces to: This is the criterion against which we will check all of the flow fields. a) u(x,y,t> = 2ox2+y2—x2~y v(x,y,t) = x3 +x(y2—4-y) 9u<x,y,t) = 4-x— 2-x-y Elwyn) = way—4) Hence 9-u + a—vssfl 6x 6y b 2 2 . 2 a a ) u(x,y,t) = My — x -y v(x,y,t) = 2-X'y- y + X —u(x,y,t) = 2'y - 2-X-y —-v(x,y,t) = 2'x- 2y ' fix 6y Hence 53—u + £9—v at 0 6X 6y 2 2 a a c) u(xsy:t) = X 't+ V(XJY>t) = X't _Y't _u(xzy,t) = 2't'x _V(X:YJt) = _t Hence 59—u + 6—v ¢ 0 6x 6y d) u(x,y,t) = (2'X+-4'Y)'X't v(x,y,t) = -3‘(x+ y)-y't a—u(x,y,t) = t-(Z-x+ 4'y) + 2-t-X a—v(x,y,t) = —t-(3-x+ 3-y) — 3't-y 6x 53' Hence ' 6—u + 8—v eé 0 3X ay NOTINCOMPRESSIBLE Problem 5.2 5 Given: Velocity fields Find: Solution: Governing Equation: Assumption: This is the criterion against which we will check all of the flow 1e a) u(x,y,z,t) = 2'y2 + 2-x.z a—u(x,y,z,t) = 2-2 fix Hence b) u(X,y,z,t) = x-y-z-t a—u(x,y,z,t) = t-y-z 6X Hence 0) u(x,y,z,t) = x2 + 2~y + 22 a—u(x,y,z,t) = 2-): Hence 5.12- Which of Efollowing sets oflequations representvpose :sible'threeadimemional incompressible flow leases-‘2. (a) u = 2}? + 2x2; 1) = —2yz +’ fifyz; w = 23x26 +55!“ (b), n==xzz1§v =>-xyzt“z; w‘= 22613—3”). a -- (C) u =2? + 2y + z“; u-=" x - 2y .‘i’ Z; W =' ail-+1?- +2.5 Which are 3D incompressible v(x,y,z,t) = —2'y~z + 6-x2-y-z 9—V(x,y,z,t) = 6'X2'Z — 22 6y 2-u+6—v+Q-w:r&0 6X By 62 V(x,y,z,t) = —x-y-z-t2 2—v(x,y,z,t) = —t2-x-z 6y iu+a—V+1W¢0 6x By 62 Iv(x,y,z,t) = x — 2~y + z a—-v(x,y,z,t) = —2 5y a—u+a—v+a—W=0 6x 6y az .INCOMPRE‘SSIBLE We will check these flow fields against the continuity equation w(x,y,z,t) = 3-)(2-22 + x3vy4 —a—w(x,y,z,t) = 6-x2-z W(x,y,z,t) = —2-X~z + y2 + 2-2 a—w(x,y,z,t) = 2 — 2-x Bz [Difficulty: 2] Problem 5.4 [Difficulty: 1] 5:4 The three components of'velocity in aveloeity field are given by u.‘=A:::.+ By+ Cz, vi: Dxrl— Ey+ Fz, and w=Gx + Hy +15, 'Detenninejthe relationship among the ,gzcoefi‘idents through I. that is necessary if. this is robe v-a pmsib‘le incompressible, flow field, Given: The velocity field provided above Find: The conditions under which this fields could represent incompressible flow Solution: We will check this flow field against the continuity equation Governing a a 6 5p . E i t' : — + — + — + — = 0 (Continuity eq ation qua Ions 64m) é1y(pv) 6 (PW) _ u ) Assumptions: (1)VIncompressible flow (p is constant) 5 Problem 5.6 Lnifficuuy: 2] fl\ Y 5.671.110 x vc'ompom’mt of. velOcity in a steade incompressible flow field in the .ryplane is u =A/x, where A.- =2"m2/s, andx isxmeasured in meters. Find the simplest y Component of velocity fer th's flow: field. Given: The x—component of velocity in a steady, incompressible flow field Find: The simplest y—component of velocity for this flow field Solution: We will check this flow field against the continuity equation Governing a a a a . . . Equations: — + — (pv) + + —’0 = 0 (Contlnmty equat1on) 6x 62 62‘ Assumptions: Incompressible flow (p is constant) @ a , (2) Two dimensional flow (velocity is not a function of z) ' \ ’ a‘u av _ ' \‘____ Based on the' two assumptions listed above, the continuity equation reduces to: — — _ 6a A — = —— There re~from continuity, we hav The partial of u with respect to X i: 6x )52 A A-y —d +fx=—+fx 2 y () x2 () Integrating this expression will yield the y-component of velocity: ' x The simplest version of this velocity component would result when f(x) = 0: Problem 5.7 [Difficulty: 2] ' A 5.7-“The y. component ofveloc'ity in asteadyanoxnpx-wsible flow field; in the .ry plane 'is v=Axy(xz—yz), were A = 3 11145“ and x and y are measured. in meters. Find the ‘ simplest .tf component 'of‘velocity for this flowfi‘eld. Given: y component of velocity Find: x component for incompressible flow; Simplest x components? Solution: Basic a_(p.u) + a_(p.v) + 2mm + ip = 0 equation: 5" 53’ 62 at Assumptions: Incompressible flow (p is constant) @ Flow is only in the x—y plane Hence 3 x — 3-x-y2) dx = —-i--A~x4 + E-A-xz-yz + f(y) Integrating This basic equation is valid for steady and unsteady flow (t is not explicit) There are an infinite number of solutions, since f(y) can be any function of y. The simplest is f(y) = 0 3 1 u(x,y) = EvA-xz-y2 — Z-A‘x4 ...
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