Question

Now evaluate the mass and momentum into and out of the CV shown with 1.0s y Rs 1.5 at (2) Let p 1200 kg/m2, Uoo- 20 m/s and cylinder radius R 0.01 m 1 cm and Az 1 m Note: The flow does not cross streamlines, so there is no flow across the side boundaries. Exit (2) NO SCALE Variable u vs y at x2-0 Inlet (1) y- H1 and v 0 constant u Uo constant v0 A) Find mass flowrate in & mass flowrate out with 0sys a (1) and 1s y/Rs 1.5 at (2) Notice that x-0 at (2) and θ-π/2 and 2θ-π , so Vr-v-0 and u--Vo Show Solve And (kg/s) (kg/s) with H = u dAINET B) Also use the stream function method to find mN &ouT&H (ans: 200 kg/s, 0.833 cm C) Find values of x-momentum in & x-momentum out. x on CV (kN) (ans: 400 kN) (kN) realistic ideal flow Consider the following 2-D, steady flowfield about a cylinder With the following Model Assumptions The upstream flow is uniform with no rotation ( ω 0 ) The fluid is incompressible ( constant ) Frictional effects are negligible, inviscid flow (H 0) This an example of "ideal flow or "potential flow This pattern is generally not achieved in nature, most notable is the absence of flow separation and a broad wake region. However the flow region upstream of the body is a close approximation of actual flows The velocity may be expressed in either rectangular (x.y) or cylindrical (r.8) coordinates with cylinder radius R. breaks down not SSSF (x,y)-( r cos(6) , r sin(θ) ) u Vr cos(e)-Vo sin(e) V.-u cos() + v sin(9) -Vr sin()V cos(6) -u sin(e) v cos(0) Here the expressions for the velocity components are known: uUse -U.(R/r) cos(20) V-U(Rir)2 sin(20) Example with freestream speed, U 20 m/s at point (x.y) (-1.2 R, 0.5 R) thus (r.q) (1.3 R, 2.7468 rad) V.--U00 [1 + (Ar)2] sin(θ)--12.2440 mis W-U00 [1-(R r) Os(6)-7.5376 m/s u-Uo-Uoo(R/r) cos(20) 11.6670 m/s --U(Rr) sin(28) 8,4031 Note: Control Volume follows boundary streamlines. Do repeat the stream-function solution. Use the given stream- and velocity field to finish the flowfield analysis. -0.5 Distance scaled by cylinder radius Steady, 2-D incompressible, inviscid, irrotational "potential flow" above a horizontal plate with a semi- circular bump obstructing the otherwise uniform horizontal flow. This is the same as the upper left quadrant for ideal frictionless flow over a circular cylinder. Steady, 2-D incompressible, inviscid, irrotational "potential flow" for the stagnation point flow above a horizontal plate. A unifom vertically downward flow approaches the stagnation point (at x#0) on horizontal plate. However an extemal source of fluid is injected into the flow at the origin producing a barrier of surface fluid that prevents the external fluid from impinging on the wall. Such a fluid barrier might help to prevent a detrimental process (erosion, corrosion, intense heat transfer, etc.) Notice the convective acceleration terms are not zero. Instead of attempting the almost impossible task of addressing the x & y momentum equations and the continuity (mass balance) equations directly the kinematic constraints div(V)=0 (continuity for constant density) and curl(V)-0 (irrotational flow due to no friction) lead to a much simpler single PDE for the stream-function or the potential function. Here, for 2-D flow with known boundaries it is best to use the stream-function solution.

Answer 1

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