CEE 3310 Dimensional Analysis & Similitude, Oct., 018 115 5.5 Review vorticity twice the local rotation rate: ω x = w y v z, ω y = u z w x, ω z = v x u y Dimensional Analysis Buckingham Pi Theorem: k = n r where k is the number of dimensionless numbers (Π s), n is the number of dimensional variables, and r is the number of physical dimensions. Found the above relationship two ways - by inspection and by a formal Buckingham Pi analysis. 5.6 Similitude & Experiments An experimenter seeks similarity between model and prototype where prototype indicates the full scale version of the item under study. We say we have total similarity if and only if all of the dimensionless parameters have the same value (at model and prototype scale). E.g., if Π 1 = g (Π, Π 3,..., Π n ) Then we have total similarity only if Π 1m = Π 1p, Π m = Π p, Π 3m = Π 3p,..., Π nm = Π np where the subscripts m and p indicate model and prototype, respectively. It turns out this is not as easy to achieve as one would like and hence rarely works out. Thus we often speak of different types of similarity, all subsets of total similarity.
CEE 3310 Dimensional Analysis & Similitude, Oct., 018 116 5.7 Types of Similarity Total Similarity = Geometric Similarity + Kinematic Similarity + Dynamic Similarity Geometric Similarity Two conditions are said to be geometrically similar if all of their length scale ratios are exactly the same. For example, in a 1/10 th scale model all of the model lengths must be 1/10 th of the prototype lengths. This includes the obvious lengths such as heights and widths but also lengths such as the radii of curvature and the roughness lengths (e.g., the average height of the unevenness of the surfaces). This last aspect can be challenging to ensure. Note that if the length ratios are all preserved then by default all of the angles are preserved between the model and the prototype. Kinematic Similarity Two conditions are said to be kinematically similar if all of their velocity scale ratios are exactly the same. Thus in addition to the length scale ratios being the same the time scale ratios must be the same too (since if all the length and time scale ratios are the same we are assured that the velocity scale ratios are all the same). If the flow can be considered frictionless (inviscid) then kinematic similarity can be achieved independently of dynamic similarity. However, if viscous forces are important dynamic similarity is required to achieve kinematic similarity. Dynamic Similarity Two conditions are said to be dynamically similar if all of their force scale ratios are exactly the same.
CEE 3310 Dimensional Analysis & Similitude, Oct., 018 117 5.8 Example Free Surface Flow Consider modeling a river in the laboratory. Our picture might look like: Based on our engineering intuition we have found that in dimensional form we expect F = f(h, V, ρ, µ, g, ɛ, σ) where ɛ is a typical height of the bottom roughness of the river and σ is the surface tension. We have k = 8 and r = 3 and thus n = 5. By inspection we see immediately that one of the Π s will be the Reynolds Number, one will be a drag coefficient C D = F/( 1ρV H ) (similar to our copepod problem but the factor of 1 shows up for historical reasons), one will simply be ɛ/h, leaving us with just two more dimensionless numbers. We see that gravity has not yet shown up so let s nondimensionalize this. Gravity is always important in flows with free surfaces because it is the restoring force for any deformation of the free surface resulting in waves (known formally as gravity waves) propagating on the free surface. For long waves (where λ, the wavelength, is much greater than the depth, the speed of propagation of a gravity wave is c = gh where H is the flow depth. So we can define a new number, like the Mach number which is: Fr = V gh known as the Froude number Which, from laboratory #3 you are now familiar with! We could have found this from
CEE 3310 Dimensional Analysis & Similitude, Oct., 018 118 our standard scaling variables too, of course. This leaves us with a final dimensionless parameter that must involve σ. Nondimensionalizing σ with our standard scaling variables (ρ, V, H) leads to the Weber number, We=ρV L/σ. Thus we have: C D = g(re, Fr, ɛ/h, We) For total similarity (and hence to insure that C Dm = C Dp ) we must require that the four dimensionless quantities in our function g all have the same values in the model and prototype. Let s start with Re and Fr. Therefore we have: and Re m = Re p = L mu m ν m Fr m = Fr p = U m glm = = L pu p ν p U p glp Therefore we can write (equating the two velocity ratios): U m = L p ν m L m ν m = = U p L m ν p L p ν p U m U p U m U p = = L p L m ν m ν p ( Lm L p L m L p Now, let s say we want to construct a 1/10 th scale model L m /L p = 1/10. Then if we are to ensure that the Reynolds number and Froude number are similar we need to work with a fluid in the model that has kinematic viscosity (0.1) 3/ ν p = 0.03ν p or in other words the kinematic viscosity of the model fluid must be just 3% of the kinematic viscosity of the prototype fluid - water in this case. If you look at appendix B in your text you will see that the only fluid with a kinematic viscosity significantly less than that of water is mercury, and the kinematic viscosity of mercury is about 10% that of water. If our model scale is 1/100 th then we need a fluid with viscosity 0.1% that of water - none exist! Thus we conclude without even looking at the remaining two nondimensional terms that total similarity can not be maintained. ) 3 In free surface flows we general hold Froude similarity and let the Reynolds number similarity be broken. In the present example if we stay with water as the model fluid and hold the Froude number scaling we find that Re m Re p = ( Lm L p ) 3
CEE 3310 Dimensional Analysis & Similitude, Oct., 018 119 and thus for 1/10 th and 1/100 th scale models we have model Reynolds numbers that are 3% and 0.1% of the prototype scale, respectively. How do we handle this? In many cases we can proceed without complete similarity (referred to as a distorted model in your text). In particular, we can accept Reynolds number dissimilarity if both Reynolds numbers are high. This is because if the Reynolds number is large enough then viscous forces become unimportant. The question is how high is high? This must be answered with a model study. Typically a series of experiments is run to study the effect of increasing Reynolds number (usually by varying the velocity) on a dimensionless parameter of interest (C D in this case) to determine when the parameter s value becomes independent of Reynolds number. 5.9 A Second Example - Drag on a Sphere and a Cylinder As we have already found from our copepod example the drag on a submerged object is a function of Reynolds number. If we are to extend this analysis to general submerged objects we need to include the roughness (we really needed roughness for the copepod too if the appendages grow and/or present themselves with different length scale ratios with respect to the original characteristic length scale of the copepod s size), hence we have: ( C D = g Re, ɛ ) D First let s consider a sphere. As already discussed the L term is typically taken as an area. Hence for a sphere we take L = πd /4 where D is the diameter of the sphere and define C D = F 1 ρπv D /4 = 8F πρv D Where F is the drag force and the factor of 1/ is included for historical reasons (Because the dynamic pressure in the Bernoulli equation has a factor of 1 in it). Hence, if we build
CEE 3310 Dimensional Analysis & Similitude, Oct., 018 10 a model and enforce geometric similarity (so that the roughness is scaled appropriately as well as the diameter) then we have: 8F m πρ m V md m = 8F p πρ p V p D p ( ) ( ) ρ p Vp Dp F p = F m ρ m V m D m We see that to ensure Reynolds number similarity and hence total similarity for this problem, and assuming we run the model tests in the same fluid as the prototype, we require Re m = Re p = L mu m ν = L pu p ν U m = U p L p L m Thus if the scale gets very small we will need to run the model at very high velocity - this may not be possible or may lead to compressibility effects (air flows) or cavitation (water flows). Now let s look at some data for two cases - C D = g(re) for both a cylinder and a sphere. For a cylinder we take the area to be the projected area of the cylinder which is simply a rectangle LD where L is the length of the cylinder and D is the diameter. Therefore we have C D = F ρv LD We will continue to define the Re based on the diameter (since we have two length scales we have to be explicit about which we are using). This is often written Re D. Finally, since we have more than one length scale in the problem, the ratio of these length scales is a dimensionless parameter that enters the problem hence we have ( C D = h Re D, ɛ D, L ) D Borrowing Fig 5.3 from our textbook (shown on next page) we see several interesting features. Clearly the functions g and h for spheres and cylinders are different. C D in each case drops with increasing Re and reaches a near constant value at 10 4 < Re < 10 5 shortly after which a sudden drop occurs (the flow is transitioning from laminar to turbulent here).
CEE 3310 Dimensional Analysis & Similitude, Oct., 018 11 Clearly ɛ/d plays a role in determining C D. L/D is important but as L D the effect on C D asymptotes toward a constant value and the physics becomes independent of L/D.
CEE 3310 Dimensional Analysis & Similitude, Oct., 018 1 5.10 Summary of Dimensionless Parameters Here are some of the most common dimensionless numbers that show up in fluid mechanics: Re forces. (Reynolds Number) ρv L µ = V L ν The ratio of intertial forces to frictional Fr (Froude Number) V gl The ratio of intertial forces to gravitational forces We (Weber Number) ρv L σ Eu (Euler Number) The ratio of intertial forces to surface tension forces. P The ratio of pressure forces to intertial forces. ρv 1 C D forces. (Drag Coefficient) F D 1 ρv A The ratio of drag forces to dynamic pressure St (Strouhal Number) fl The ratio of event frequency (often vortex shedding) V and the advective frequency (inverse of the advective, or inertial, time scale). V Ma (Mach Number) The square root of the ratio of intertial forces to c compressibility effect forces can be thought of as a Froude number for compressible flows.