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Assignment Task

Part one: Simple initial value problem

Assignment (please read the problem description before doing the assignment)

• Find out the equation of motion of the cylinder. The derivation procedure of the equation must be included in the
• Develop a numerical method for predicting the vibration of the cylinder. The procedure of deriving the numerical formula must be included in the
• Develop a MATLAB program for solving the problem. The MATLAB program code must be included in the
• Keeping the damping coefficient of c = 100 Nꞏm/s, conduct numerical simulations for KC=2 to 20 with an interval of 2 and find out the variation of the power with the KC
• Keeping the KC number to be constant of KC=10, conduct numerical simulations for c=0 to 2000 Nꞏm/s with an interval of 50 and find the variation of the power with
• In the report, show the time histories of the vibration for all the calculated
• Discuss how KC and c affects the power

Problem description

An energy extraction device includes a cylinder elastically mounted on a spring and the electricity generator. The cylinder vibrates only in a direction parallel to the flow and drives an electricity generator. The electricity generator extracts the energy from the cylinder in the same way as a damper. It is modelled as a damper with a damping constant c when the vibration is studied. The flow velocity is a combined steady and oscillatory flow

u ( t ) = U 0 + U m sin(2p t / T )

where U 0 is the steady flow velocity, U m is the amplitude of the oscillatory flow velocity, T is the period of the oscillatory flow and t is the time.

The KC number is defined as

The electricity generator can modelled by a damper that extract energy from the motion of the cylinder with a constant damping constant of c. The structural damping is considered to be negligibly small. Considering the power given by the fluid should be the same as the power received by the damper, the power can also be calculated using the energy received by the electricity generator (damper) as.

where P is the power of the system, F is the hydrodynamic force on the cylinder, V is the vibration speed of the cylinder. In the numerical simulation, if total computational time step is N , the power can be calculated by

Note, N0 must be a step where the vibration has been fully developed. In addition, the time from step N0 to N0+N must be one or a few whole periods.

The hydrodynamic force on the cylinder can be predicted by the Morison equation

where V r= u – V is the velocity of the flow relative to the cylinder, C A and C D and inertia and drag coefficients, respectively, m d is the displaced fluid mass and A p is the projected area. The calculating parameters are listed in the table below.

Part two: One-dimensional convection diffusion problem

Assignment (please read the problem description before doing the assignment)

• Develop a finite difference method (FDM) formula for solving the The procedure of deriving the FDM formula must be included in the report.
• Develop a MATLAB program and use this program to do the simulation and answer the following Divide the soil depth into 200 cells when solving the problem numerically. You must attach the MATLAB program in your report.
• If α =0.0002 m ² /s, find out when the temperature at the depth of 5 m reaches 25°C.
• If the temperature at the depth of 1.5 m reaches 25°C is defined as T5, do the simulations for α in the range of 0.0002 m ² /s to 0.001 m ² /s with an interval of 0.0001 m ² /s and discuss how α on T1.5 and why.

Problem description

Note: In the paragraph below, the number j stands for the last digit of your student ID. For example, if your student ID is 2345673, 10+j = 13.

The water on the ground flows into the soil through the pore space of the soil (the space between soil particles). The flow velocity is very small of 0.002×(1+j) m/s and it does not change along the vertical direction. Initially, the temperature is (10+j) °C in the whole soil volume (including soil and the water in the soil). The temperature of the water on top of the ground level is 30°C and remains constant. The warm water moves into the soil and increases the temperature in the soil.

A coordinate x is defined and its origin is on the ground level and pointing downwards. The governing equation is the convection diffusion equation

Where

• T is the temperature in the soil
• t is the time
• u is the water flow velocity
• α is the heat diffusion coefficient (the influence of the soil has been considered).

Boundary condition is T =30°C at the top boundary and 0 at the bottom boundary (at x = 2 m).