Mech 5306 - construct the velocity gradient matrix for flow

Question

Problem 1

Flow around a 60° corner can be expressed by the following stream function:

ψ = C(x²y - y³/3)

Where C is a constant.

a) Plot the stream function for several streamline values and show the direction of the flow (assume C = 1 for your plots)

b) Use Bernoulli's equation to determine the pressure distribution, i.e. p(x, y). Neglect changes in elevation for this problem

c) Construct the velocity gradient matrix for this flow

Problem 2

Consider the following flow field:

u = x/(1+t)    v = y/(1 +2t)

a) Find the acceleration of the flow field

b) Determine the stream function and plot representative streamlines for t = 0

Problem 3

Consider the following flow field for an incompressible fluid:

u = -6x^i   v = 6y^?

a) Determine if the flow field satisfies the conservation of mass

b) Find the rotation of a fluid element located at point (1,2)

c) Assuming that the pressure at point (1,2) is 173kPa find the pressure at point (1,1) assuming a fluid density of 1200 kg/m³.

Problem 4

We'll use streamlines to help us describe water flow through a small channel. Using the below stream function, determine the following:

ψ = 2x² - 2y²

1) If the equation satisfies the conservation of mass for incompressible flow

2) Determine the velocity gradient matrix for this flow

3) Plot the stream function for several streamlines

4) If the pressure at point A (1.5, 1.5) is 101325 Pa, what is the pressure at point B (0.5, 0). Take the density of water to be 1000 kg/m³

Problem 5

At UTEP, we are in the initial phases of designing a CubeSat propulsion system. One of the things we would like to know is how much combustion gases we should expel from the nozzle of the propulsion system to obtain a certain ΔV (pronounced delta V). This ΔV represents the change in velocity before and after the propulsion system firing.

If the CubeSat weighs 50 kg and the mass of the gases expelled from the system is 10 kg in 3 seconds and the velocity of these gases are at 300m/s relative to a direction opposite to the satellite's trajectory. Assume that the firing of this satellite is done in space (no gravity or drag forces) and that it is steady. Determine, the using the relationships for conservation of momentum:

1) Thrust in N
2) The acceleration of the cubesat during the 3 second burn time
3) The ΔV of the cubesat during the 3 second burn time