The Role of Modelling in Engineering Design. John Fitzpatrick Professor of Mechanical Engineering Trinity College Dublin
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1 The Role of Modelling in Engineering Design John Fitzpatrick Professor of Mechanical Engineering Trinity College Dublin
2 Equation of Fluid Motion (Navier-Stokes Equations) = z v y v x v y p z v w y v v x v u t v ν ρ = z u y u x u x p z u w y u v x u u t u ν ρ = z w y w x w z p z w w y w v x w u t w ν ρ = z w y v x u
3 Navier Stokes Equations No general solution exists Computational methods (CFD) Experiments required How do we conduct the experiments?
4 CFD Study of a Jet
5 Modelling Most experiments conducted for design purposes involve modelling. The model is often either smaller (or bigger) than the real construction. How do we conduct model tests and relate the results to the real thing (i.e. the prototype)
6 The A380 Paradox Aerospatiale proposal 1994 How do we build an aircraft twice the size of A340 & 747? (Boeing 747 & Airbus 340 aircraft were considered to be at the limit of aircraft size.)
7 The Design Challenge Doubling the size has the following consequences : a. The mass increases by a factor of 8. b. The lift increases by a factor of 4. The Challenge a. Increase the lift effectiveness b. Reduce the weight c. Do both The Answer The first A380 was delivered last year to Singapore Airlines (18 months late!)
8 Fundamentals force = mass x acceleration (ma) work = force x distance (Fs) energy== work power = rate of work (FU)
9 Basic Dimensions Space (m) Time (s) Mass (kg) Temperature (K)
10 Derived Units Force (1N=1kgm/s 2 ) Work (1J=1Nm=1kgm 2 /s 2 ) Energy (J) Power (1W=1J/s)
11 Functional Relationships These are expressions in which a dependent variable is writen as function of the other independent variables which affect the dependent variable.
12 A functional relationship The force (F) required to move a body through a fluid can be expressed as : F=f(d,L,U,ρ,µ) Where d = diameter L = length U = velocity ρ = density µ = viscosity (friction)
13 Important Parameters Geometric Length, Area, Volume Kinematic/Dynamic Variables Velocity, Time, Temperature Forces, Pressure, Stress, Strain Properties Density (ρ) Viscosity (µ) Elastic modulus Specific heat, conductivity etc.
14 How to design a Model Test Geometric Scaling Force Scaling We are going to consider applications in Fluid Flows
15 A Question A series of tests are conducted on a model of a windmill which is 1/5 th the size of the proposed system (prototype) to determine the power output. What do you think the power output for the full size (prototype) windmill will be? 5 times, 10 times, 100 times! (We will return to this at the end)
16 Geometric Scaling Relatively easy to achieve as all geometric attributes can readily be scaled and a physical model constructed.
17 Force Scaling This is more conceptually more difficult, but is relatively straightforward. Unfortunately, there are a number of applications where we cannot scale all the important forces in the same model test.
18 Forces in Fluid Systems Inertial Forces :these are due to the mass (density) of the moving fluid. Friction Forces : these are due to the viscosity of the fluid which resists motion. Gravity Forces : these are due to gravitational action and arise in water type flows (e.g. waves). Elastic Forces : these are due to compressibility effects and are important at very high speeds.
19 Aerospace Systems Key parameters Geometric length & width L & d Kinematic/ Flow Variable speed U Fluid Properties density ρ viscosity µ
20 A Functional Relationship F=f(d,L,U,ρ,µ) F is dependent on 5 variables. To determine the nature of each dependency, it would be necessary to conduct tests in which all the parameters but one are kept constant so that the dependence of F on this variable is established. This would then be repeated for the remaining four.
21 Dimensonal Analysis It is possible to regroup the 5 dimensional variables into 3 dimensionless groups as follows. e.g. F L,, ρud ρ U 2 d 2 d µ
22 Dimensionless Groups These groups represent the following : Required Force/Inertia Force Length/Diameter Inertia Force/Viscous(Friction) Force The Inertia Force (~ρu 2 ) is the most important in fluid flows and groups are usually formed as the ratio of this to other forces acting.
23 Dimensionless Relationship We can now write a dimensionless functional form as : F L ρ Ud ρ U 2 d 2 = φ ( d, µ ) So that we can now investigate the forces required to move bodies with different slenderness ratios (L/d) through a fluid
24 Modelling Using dimensionless relationships enables us to perform model tests. The key is that in addition to having a geometric model, the force ratios must be scaled properly.
25 Example A series of tests are to be conducted on a model car of 1/5 scale to investigate the drag force resulting from posible design modifications. If the prototype speed is 160 km/hr, at what speed should the test be conducted? It the results show that the drag force is 1.75kN, what is the drag on the protype?
26 A geometric scale model is constructed to 1/5 scale. i.e. (L/d) prot =(L/d) model Assume the density and viscosity of the air are the same for the test and full scale conditions. U prot =45m/s then so that (ρud/µ) prot = (ρud/µ) model U model =225m/s
27 The force on the prototype (full scale) is then determined from (F/ρd 2 U 2 ) prot = (F/ρd 2 U 2 ) model so that F prot =1.75kN and the power can be obtained as Power = Force x Velocity so that Power = 78.75kW
28 Some Limitations Some dimensionless groups will not scale simultaneously. For boats, viscous, gravity and inertia forces are important. inertia/viscous = (ρud/µ) inertia/gravity = (ρu 2 d 2 /ρgd 3 ) = (U 2 /gd) (Consider a 1/10 th scale model!)
29 Consequences Separate tests are conducted to determine a. the power required to overcome the gravity (wave) forces b. the power required to overcome the viscous forces The results are then combined to determine the overall power required for the vessel.
30 Further Problems! Scaling aircraft at cruise conditions with U=250m/s (900km/hr, 560mph). Using a wind tunnel atmospheric conditions gives 1/10 th scale model testing velocities at ~2000m/s. This is equivalent to 6 times the speed of sound (M=6) and means the elastic or compressible forces become important.
31 Wright Brothers
32 Wright Flyer
33 German/Dutch Low Speed Wind Tunnel Capacity for Testing both full size & scale models Test Section Dimensions 9.5m x 9.5m Flow Velocity 80 m/s Aircraft models are tested for take off and landing conditions.
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42 The Propeller/Windmill Parameters Power (dependent) Diameter (d) & Chord width (c) Rotational Speed (N) Wind Speed (U) Density (ρ) i.e. Power, W = f(d,c,n,u,ρ)
43 Scaling for Props/Windmills The groups for scaling are: c/d U/Nd & W/ρN 3 d 5
44 Scaling for Windmills The last grouping shows that the power scales with the 5 th power of diameter! Thus, a full size windmill which rotates at 1/5 th of the model can generate 3125 times that of a 1/5 th scale model!
45 Thank you for your attention Any Questions?
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