DEPARTMENT OF NAVAL ARCHITECTURE &

MARINE ENGINEERING

Degree of BEng in Naval Architecture & Ocean Engineering Degree of BEng in Naval Architecture & Small Craft

Degree of MSc/PGDip Offshore Floating Systems

NM422 Numerical Methods for Naval Architects NM402 Theory & Practice of Marine CFD

NM915 Theory & Practice of CFD

1st April 2010 9.30-11.00

ANSWER TWO QUESTIONS

(All Questions worth equal marks)

Calculators must not be used to store text and/or formulae nor be capable of communication. Invigilators may require calculators to be

reset.

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NM422 Numerical Methods for Naval Architects - 1st April 2010 NM402 Theory & Practice of Marine CFD NM915 Theory & Practice of CFD 1. (a) By considering a one-dimensional flow domain with unequal-sized cells,

explain with the aid of sketches the following schemes for estimating cell-face-average values of flow parameters: (i) central differencing (ii) first-order upwinding (iii) second-order upwinding In each case write down equations for the reconstructed cell-face-average values of in terms of the cell-centre values, and discuss the advantages and disadvantages of each approach. How would you estimate the cell-face average gradients of ? (14)

(b) For the one-dimensional flow case shown below:

P Q R

6.300 5.670 5.103

(c) calculate estimates for the cell-face average value of at the face between Q

and R using each of these three approaches, using the following information: Cell length cell

centrexcell

centrecell

centrevelocity

cellcentregradient

(mm) (mm) (mm/s) (1/mm)

P 6.300 34.150 14.000 10.000 0.317

Q 5.670 40.135 17.000 10.000 0.794

R 5.103 45.522 23.000 10.000 1.274(6)

(d) Explain how the third order QUICK scheme attempts to improve on these schemes, and explain one advantage and one disadvantage of this approach. (5)

Contd./... Page 2 of 4

NM422 Numerical Methods for Naval Architects - 1st April 2010 NM402 Theory & Practice of Marine CFD NM915 Theory & Practice of CFD

2. (a) Explain briefly what is meant by a Newtonian fluid. Given one

example of a Newtonian fluid and a non-Newtonian fluid. (4)

(b) The Navier-Stokes equations in finite volume conservation form are written as:

( ) V S V S S

dV dS dV p T dSt

+ = + V V V.n b dS n

What fundamental physical principle is expressed by this equation? What is the meaning of each side of the equation?

Explain the physical meaning of each of the five terms.

(8)

(c) What are the flow variables we wish to calculate with this equation and what additional equation(s) would we typically need for the solution of:

(i) an incompressible fluid, (ii) a compressible fluid and

what physical principles do these additional equations state?

(6) (d) The approximated finite volume form of the convection diffusion

equation for the variable is written as:

cc f f

f f f

V Ft n + =

fS

Explain clearly the meaning of the parameters c , , cV f , fF , fS and

fn

in this equation.

(3) (d) Explain how the second term in the equation above is obtained from

the equivalent term in the exact finite volume form of the convection-diffusion equation:

V S S

dV dS dSt n

+ = V.n

(4) Contd./...

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NM422 Numerical Methods for Naval Architects - 1st April 2010 NM402 Theory & Practice of Marine CFD NM915 Theory & Practice of CFD 3. (a) Describe with the aid of sketches the following types of CFD grids,

explaining their key features, and their advantages and disadvantages compared to other grid types:

(i) structured grid

(ii) block-structured grid

(iii) unstructured grid

(16)

(b) Explain briefly with the aid of sketches why the use of skewed or high- aspect ratio elements can lead to inaccuracy in the reconstruction of cell-face-averaged convective and diffusive fluxes.

(5)

(c) Explain with the aid of examples what is meant by modelling error, how it

arises, and how its magnitude may be evaluated. (4)

END OF PAPER

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DEPARTMENT OFDegree of BEng in Naval Architecture & Small Craft