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Repeat Example 6.1, but with the somewhat more realistic assumption that the flow is similar to a free vortex (irrotational) profile, Vθ = c/r (where c is a constant), as shown in Fig. P6.25. In doing so, prove that the flow rate is given by
and w is the depth of the bend.
P6.25
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Fox and McDonald's Introduction to Fluid Mechanics
- Example(1) Consider the water siphon shown in Fig. below. Assuming that Bernoulli's equation is valid, (a) find an expression for the velocity V2 exiting the siphon tube. (b) If the tube is 1 cm in diameter and z = 60 cm, z2 =25 cm, estimate the flow rate in m²/s. Estimate the pressure at point(3) if 23=90cw z =0---- V2 НОНЕWORК Estimate the pressure at point (3 ) if Z3= 90cm.arrow_forwardQ.4 A steady, uniform-density, 2-D flow is to be calculated on the square grid shown below. The boundary velocities are given as; v₁ =30, V = 40,uc=100, u = 50, u = 200, u, = 210, V = 0 and v₁ = 20. Among these numbers, there is some doubt about correctness of the value of u,. If all other numbers are correct, what should be the correct value of u,? The internal velocities are governed by simplified momentum equations given by: up = 70+0.5 (P₁-P₂) u, = 10 +0.7 (P3-P4) V =30+0.5(P3-P₁) VG =18+0.8(P₁-P₂) Write discretized continuity equation for each control volume. Derive the discretization equation for pressure by substituting from momentum equations, following SIMPLER calculation procedure. Solve the pressure equations to obtain P₁, P2, P3 and P₁. Hence obtain values of up, u, V and VGarrow_forward1. For a certain incompressible two-dimensional flow, the stream function, ψ(x, y) is prescribed. Is the continuity equation satisfied? 2. If u = −Ae−ky cos kx and v = −Ae−ky sin kx, find the stream function. Is this flow rotational, or irrotational?arrow_forward
- Q.4 A steady, uniform-density, 2-D flow is to be calculated on the square grid shown below. The boundary velocities are given as; v₁ = 30, V = 40,uc=100, u = 50, u = 200, u, = 210, v = 0 and v₁ = 20. Among these numbers, there is some doubt about correctness of the value of u,. If all other numbers are correct, what should be the correct value of u,? The internal velocities are governed by simplified momentum equations given by: up=70+0.5(P₁-P₂) u, = 10+0.7 (P3-P4) V=30+0.5(P₁-P₁) V=18+0.8(P₁-P₂) Write discretized continuity equation for each control volume. Derive the discretization equation for pressure by substituting from momentum equations, following SIMPLER calculation procedure. Solve the pressure equations to obtain P₁, P2, P3 and p₁. Hence obtain values ofu,, U₁, V and V6.arrow_forwardDetermine whether each of the followings are rotational flow or irrotational flow. Also, determine their stream functions. (u: x-component of velocity. v: y-component of velocity.) 1. u=x^3-3(x)(y^2), v=y^3-3(x^2)(y) 2. u=2xy, v=-y^2arrow_forward6.52 The motion of a liquid in an open tank is that of a combined vortex consisting of a forced vortex for 0 2 ft. The velocity profile and the corresponding shape of the free surface are shown in Fig. P6.52. The free surface at the cen- ter of the tank is a depth h below the free surface at r = o. Deter- mine the value of h. Note that h = hforeed + hiees where hforeed and hfree are the corresponding depths for the forced vortex and the free vortex, respectively. (See Section 2.12.2 for further discussion regarding the forced vortex.) 10 Ve, ft/s r, ft 2 2 r, ft h IFIGURE P6.52arrow_forward
- The streamlines in a particular two-dimensional flow field are all concentric circles, as shown in Fig. 5 the equation v, = wr where w is the angular velocity of the rotating mass of fluid. Determine the circulation around the path ABCD. The velocity is given by B Figure 5arrow_forwardThe two-dimensional incompressible flow-field around a Rankine vortex of strength r and core size R, has only a circumferential component, v. The value of v is given as follows: R г v(r) Tr r R 2r Determine the vorticity of this flow-field and the pressure coefficient, Cp, and plot them as a function of r/R. The non-dimensional pressure coefficient C, is defined below. Where is the pressure expected to be minimum, and what is the value of the minimum C,? (Use polar coordinates) p - Poo Cp where p. is the pressure at infinity, and p is the density of the fluid. VR is the flow velocity at the outer edge of the vortex core (i.e. at r= R)arrow_forwardQ.5 The velocity components in x and y direction 2 are given by u = Axy° - xy; v = > ху; v — ху = xy² – 3/4 .4 y*. The value of A for a possible flow field involving an incompressible fluid is: A -3/4 В 3 C 4/3 D -4/3arrow_forward
- Example(1) Consider the water siphon shown in Fig. below. Assuming that Bernoulli's equation is valid, (a) find an expression for the velocity V2 exiting the siphon tube. (b) If the tube is 1 cm in diameter and z1 = 60 cm, z2 =25 cm, estimate the flow rate in m'/s. Estimate the pressure at point(3)if 23=90 m っ。 z=0- -----Z2 V2 TYarrow_forward3.4 Consider a steady, incompressible, 2D velocity field for motion parallel to the X-axis with constant shear. The shear rate is du/dy Ay. Obtain an expression for the velocity field V. Calculate the rate of rotation. Evaluate the stream function %3D for this flow field. Ay Ay + В і, о, Ay + By+ C 6. Ans: V= 2arrow_forwardFor the flow defined by the stream function ψ = V0y: (a) Plot the streamlines. (b) Find the x and y components of the velocity at any point. (c) Find the volume flow rate per unit width flowing between the streamlines y = 1 and y = 2.arrow_forward
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