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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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