In the flow passage below, the velocity is varying with time. The velocity is gi

Question

In the flow passage below, the velocity is varying with time. The velocity is given by Q/A, where Q is in m3/s (Q is the flow rate) and A is the cross sectional area in m2. Q varies with time as:

Q=Q0-Q1 *t/to

At time 0.5 s, it is known that at section A-A the velocity gradient in the x-direction is 2m/(s.m). Given that Q0, Q1and t0 are constants with values of 0.985m3/s, 0.5m3/s, and 1 s, respectively, and assuming the flow is one-dimensional (varies in x direction only) calculate the velocity, local acceleration and convective acceleration at section A-A at time t=0.5s. The local channel diameter at section A-A is 50cm.

Explanation / Answer

area at A-A = pi/4 * 0.5^2 = 0.196 m^2

Q = Q0-Q1*t/t0

at t=0.5s

Q = 0.985-0.5*0.5/1 = 0.735 m^3/s

A*V = Q

V = Q/A = 0.735/0.196 = 3.75 m/s

V = Q/A = (Q0-Q1*t/t0)/A

local acceleration, a = del(V)/del(t) + V*del(V)/del(x) = -Q1/(t0*A) + V* del(V)/del(x)

at given section

a = -0.5/(1*0.196) + 3.75*2 = 4.949 m/s^2

convective acceleration = V*del(V)/del(x) = 3.75*2 = 7.5 m/s^2

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