Scaling and Dimensional Analysis

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Horace Sharp
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1 Scaling and Dimensional Analysis Understanding the role of dimensions and the use of dimensionless numbers for evaluating the forces in a system so that the correct assumptions about it can be made and can be modelled effectively. Resources Massey, prologue, chapter 9 White, chapter 5 Alexandrou, chapter 6.

2 Why use dimensional analysis? When starting to look at real systems, it is often not possible to say which assumptions apply straight away. Dimensional analysis allows analysis of a system when very little is known about the physics acting within it. Dimensional analysis allows the effects of different forces to be properly evaluated on the behaviour of a system and on its solution. Dimensional analysis allows measurements to be generalised from a particular set-up to the system in general.

3 Similarity & Dimensional Analysis In practical problems, often exact conservation equations are difficult to write down because not enough is known about the behaviour of the system. To make progress, it is necessary to simplify the e quations as much as possible and then use to empirical evidence. A tool for doing this is dimensional analysis: the formation of non-dimensional numbers. Its advantages include: reducing the number of variables; you do not need to understand the detailed physical mechanisms within the system; removing the effects of scale.

4 Similarity Geometric similarity: do the two systems look the same? Kinematic similarity: streamlines of a flow are geometrically similar. Kinematically similar flows must be geometrically similar, but geometrically similar flows are not necessarily kinematically similar. Dynamic similarity: same ratio between resultant forces and corresponding locations on boundaries of two systems. Dynamically similar flows must be kinematically similar and have similar mass distributions. This takes place if all the non-dimensional groups in a flow are the same in a geometrically similar flow.

5 Dimensional Analysis All equations must be dimensionally homogeneous. Dimensions are things like mass (M), length (L) and time (T). If the variables affecting a parameter are known then they can be manipulated to form dimensionless groups (called Πs ). This is useful because: it ensures dynamical similarity (so model testing can be employed); reduces the number of variables that have to be used; the groups are physically meaningful and can tell you what is happening within a flow e.g. Re=ρud/µ=inertial forces/viscous forces.

6 Buckingham s Pi theorem When a variable q 1 depends on a number of other parameters q 2, q 3,,q n or q = gq (,q,...,q ) n then they may be grouped into n-r dimensionless groups (or pis), where r is the number of independent dimensions, so that Π 1 = G(Π 2,Π 3,...,Π n r )

7 Example 1 of dimensional analysis: pressure drop of a viscous, incompressible fluid through a straight pipe Pressure drop p is a function of fluid density ρ, fluid viscosity µ, mean velocity U, length of pipe L, diameter of pipe D, surface roughness e.

8 Common non-dimensional numbers Reynolds number Re=ρul/µ Inertial/viscous forces Froude number Fr=u/ (lg) Inertial/ gravitational forces Mach number M=u/a Speed/speed of sound Weber number We=ρu 2 l/σ Inertial/ surface tension forces Euler number (e.g. Friction/drag/lift coefficients) Eu= p/ρu 2 Pressure/ inertial forces

9 Example 2 of dimensional analysis: what is the energy of an atomic blast? The change of radius with time is known (from a series of photographs in Life magazine). The radius of the blast depends on the energy of the blast E, atmospheric density ρ, and time t.

10 The use of scaling in engineering systems What are the dominant forces in a system? Which terms should be included in the momentum (and other) conservation equations? Can be found by non-dimensionalising the momentum equation (so it only includes dimensionless numbers), and estimating the size of the different terms. Example 1: Flow of a solidifying liquid that forms insulating board through a perforated distributor.

11 Schematic of apparatus Poker Mixing chamber Entry of fluids Jets Surface

12 Flow near a single hole M.A.Gilbertson, Proc. I.Mech.E. part E:Process Mechanical Engineering 216(E2) (2002) p 0 u p d h d p u h p a U = 1.7 m/s, d h = 3 mm, d p = 19mm, σ = 0.06 N/m, µ = 0.5 kg/ms, ρ = 1250 kg/m 3.

13 Result of scaling analysis Dimensionless Value Industrial Larger diameter Inertial Gravity Losses Surface T Dimensionless Groups Poker type

14 Pictures of flow in the model

15 Flow at the end of the poker

16 Example 2: Flow past a cylinder Example 6.4, p.226 Alexandrou Consider air flow past a cylinder a diameter D of 0.1m. at a velocity V of 100m/s. For this flow, establish the importance of viscous and gravitational forces relative to the inertia forces. 1. What are the forces? 2. How are the terms represented? 3. How are these terms approximated? 4. How large are the force ratios? (Dimensionless numbers) 5. Are these results right?

Dimensional and Model Analysis

Dimensional and Model Analysis 5.1 Fundamental dimensions 5.2 Rayleigh s and Buckingham s method 5.3 Dimension less numbers and their significance 5.4 Hydraulic similitude 5.5 Type of models 5.6 Distorted