Suppose we are interested in the ef

Suppose we are interested in the effects of wave action on a large ship at sea, heat loss of a submarine and the drag force it experiences in its underwater environment, or the wind effects on an aircraft wing. Quite often, because it is physically impossible to duplicate the actual phenomenon in the laboratory, we study a scaled-down model in a simulated environment to predict accurately the performance of the physical system. The actual physical system for which the predictions are to be made is called the prototype. How do we scale experiments in the laboratory to ensure that the effects observed for the model will be the same effects experienced by the prototype? Although extreme care must be exercised in using simulations, the dimensional products resulting from dimensional analysis of the problem can provide insight into how the scaling for a model should be done. The idea comes from Buckingham's theorem. If the physical system can be described by a dimensionally homogeneous equation in the variables, then it can be put into the form
f(Π_1,Π_2,…,Π_n )=0
for a complete set of dimensionless products. Assume that the independent variable of the problem appears only in the product Π_nand that
Π_n=H(Π_1,Π_2,…,Π_(n-1))
For the solution to the model and the prototype to be the same, it is sufficient that the value of all independent dimensionless products Π_1,Π_2,…,Π_(n-1 )be the same for the model and the prototype.
For example, suppose the Reynolds number υrρ/μappears as one of the dimensionless products in a fluid mechanics problem, where υ represents fluid velocity, r a characteristic dimension (such as the diameter of a sphere or the length of a ship), ρ the fluid density, and μ the fluid viscosity. These values refer to the prototype. Next, let ρ_m,υ_m,r_m and μ_(m )denote the corresponding values for the scaled-down model. For the effects on the model and the prototype to be the same, we want the two Reynolds numbers to agree so that
(υ_m r_m ρ_m)/μ_m = υrρ/μ
The last equation is referred to as a design condition to be satisfied by the model. If the length of the prototype is too large for the laboratory experiments so that we have to scale down the length of the model, say r_m=r/10, then the same Reynolds number for the model and the prototype can be achieved by using the same fluid ( ρ_m=ρ and μ_m= μ ) and varying the velocity, υ_m=10υ. If it is impractical to scale the velocity by the factor of 10,we can instead scale it by a lesser amount 0 < k < 10 and use a different fluid so that the equation
(kρ_m)/〖10μ〗_m =ρ/μ
is satisfied. We do need to be careful in generalizing the results from the scaled-down model to the prototype. Certain factors (such as surface tension) that may be negligible for the prototype may become significant for the model. Such factors would have to be taken into account before making any predictions for the prototype.
EXAMPLE 1 Drag Force on a Submarine
We are interested in the drag forces experienced by a submarine to be used for deep-sea oceanographic explorations. We assume that the variables affecting the drag D are fluid velocity υ, characteristic dimension r (here, the length of the submarine), fluid density ρ,the fluid viscosity μ, and the velocity of sound in the fluid C. We wish to predict the drag force by studying a model of the prototype. How will we scale the experiments for the model?
A major stumbling block in our problem is in describing shape factors related to the physical object being model ,in this case, the submarine. Let's consider submarines that are ellipsoidal in shape. In two dimensions, if a is the length of the major axis and b is the length of the minor axis of an ellipse, we can define r_1=a/b and assign a characteristic dimension such as r, the length of the submarine. In three dimensions, define also r_2=a/b', where a is the original major axis and b' is the second minor axis. Then r,r_1 and r_2 describe the shape of the submarine. In a more irregularly shaped object, additional shape factors would be required. The basic idea is that the object can be described using a characteristic dimension and an appropriate collection of shape factors. In the case of our three-dimensional ellipsoidal submarine, the shape factors r_1 and〖 r〗_2 are needed. These shape factors are dimensionless constants.
Returning to our list of six fluid mechanics variables D,υ,t,ρ,μ,and c, notice that we are neglecting surface tension (because it is small) and that gravity is not being considered. Thus, it is expected that a dimensionless analysis will produce three (6- 3) independent dimensionless products. We can choose the following three products for convenience:
Reynolds number R=υrρ/μ
Mach number M=υ/c
Pressure coefficient P=p/(ρυ^2 )
The added shape factors are dimensionless so that Buckingham's theorem gives the equation
h(P,M,R,r_1,r_2 )=0
Assuming that we can solve for P yield
P=H(M,R,r_1,r_2 )
Substituting P=p/〖ρυ〗^2 and solving for p gives
p=pυ^2 H(R,M,r_1,r_2)
Remembering that the total drag force is the pressure (force per unit area) times the area (which is proportional to r^2 for geometrically similar objects) and gives D∝pr^2 the proportionality, or
D=kρυ^2 r^2 H(R,M,r_1,r_2) (1)
Now a similar equation must hold to give the proportionality for the model
D_m=kρ_m υ_m^2 r_m^2 H(R,M,r_1m,r_2m) (2)
Because the prototype and model equations refer to the same physical system, both equations are identical in form. Therefore, the design conditions for the model require that
Condition (a) R_m=R
Condition (b) M_m=M
Condition (c) r_1m= r_1
Condition (d) r_2m=r_2
Note that if conditions (a)-(d) are satisfied, then Equations (1) and (2) give
D_m/D=(ρ_m υ_m^2 r_m^2)/(〖ρυ〗^2 r^2 ) (3)
Thus, D can be computed once D_m is measured. Note that the design conditions (c) and (d) imply geometric similarity between the model and the prototype submarine
a_m/b_m =a/b and a_m/〖b'〗_m =a/b'
If the velocities are small compared to the speed of sound in a fluid, then υ/c can be considered constant in accordance with condition (b). If the same fluid is used for both the model and prototype, then condition (a) is satisfied if
υ_m r_m=υr
or
υ_m/υ=r/r_m
which states that the velocity of the model must increase inversely as the scaling factor r_m/r. Under these conditions, Equation (3) yields
D_m/D=(ρ_m υ_m^2 r_m^2)/(〖ρυ〗^2 r^2 )=1
If increasing the velocity of the scaled model proves unsatisfactory in the laboratory, then a different fluid may be considered for the scaled model (ρ_m≠ρ and μ_m≠μ).If the ratio υ/c is small enough to neglect, then both υ_m and r_m can be varied to ensure that
(υ_m r_m ρ_m)/μ_m =υrρ/μ

in accordance with condition (a). Having chosen values that satisfy design condition (a), and knowing the drag on the scaled model, we can use Equation (14.30) to compute the drag on the prototype. Consider the additional difficulties if the velocities are sufficiently great that we must satisfy condition (b) as well.
A few comments are in order. One distinction between the Reynolds number and the other four numbers in fluid mechanics is that the Reynolds number contains the viscosity of the fluid. Dimensionally, the Reynolds number is proportional to the ratio of the inertia forces of an element of fluid to the viscous force acting on the fluid. In certain problems the numerical value of the Reynolds number may be significant. For example, the flow of a fluid in a pipe is virtually always parallel to the edges of the pipe (giving laminar flow) if the Reynolds number is less than 2000. Reynolds numbers in excess of 3000 almost always indicate turbulent flow. Normally, there is a critical Reynolds number between 2000 and 3000 at which the flow becomes turbulent.
The design condition (a) mentioned earlier requires the Reynolds number of the model and the prototype to be the same. This requirement precludes the possibility of laminar flow in the prototype being represented by turbulent flow in the model, and vice versa. The equality of the Reynolds number for a model and prototype is important in all problems in which viscosity plays a significant role.
The Mach number is the ratio of fluid velocity to the speed of sound in the fluid. It is generally important for problems involving objects moving with high speed in fluids, such as projectiles, high-speed aircraft, rockets, and submarines. Physically, if the Mach number is the same in model and prototype, the effect of the compressibility force in the fluid relative to the inertia force will be the same for model and prototype. This is the situation that is required by condition (b) in our example on the submarine.