1. Dimensional Analysis

2. Experimentation and modeling are widely used techniques in fluid mechanics.

3. It is important to develop a meaningful and systematic way to perform an experiment.

4. to design a pipeline, “ The pressure drop per unit length that develops along the pipe as a result of friction.” first step in the planning of an experiment to study this problem would be to decide on the factors, or variables, that will have an effect on the pressure drop per unit length

7. How many dimensionless products are required to replace the original list of variables?

9. A dimensional analysis can be performed using a series of distinct steps. Step 1. List all the variables that are involved in the problem. Step 2. Express each of the variables in terms of basic dimensions. Step 3. Determine the required number of pi terms. Step 4. Select a number of repeating variables, where the number required is equal to the number of reference dimensions Step 5. Form a pi term by multiplying one of the nonrepeating variables by the product of the repeating variables, each raised to an exponent that will make the combination dimensionless. Step 6. Repeat Step 5 for each of the remaining nonrepeating variables. Step 7. Check all the resulting pi terms to make sure they are dimensionless. Step 8. Express the final form as a relationship among the pi terms, and think about what it means

10. Step 1. List all the variables that are involved in the problem. Step 2. Express each of the variables in terms of basic dimensions.

11. Step 3. Determine the required number of pi terms. Since there are five (k=5)variables (do not forget to count the dependent variable, ) and three required reference dimensions (r=3)then according to the pi theorem there will be (5-3=2),or two pi terms required.

12. The repeating variables to be used to form the pi terms need to be selected from the list D, and V. (Remember, we do not want to use the dependent variable as one of the repeating variables.) Step 4. Select a number of repeating variables, where the number required is equal to the number of reference dimensions Three reference dimensions are required, we will need to select three repeating variables we will use D, V, and as repeating variables. we cannot form a dimensionless product from this set.

13. Step 5. Form a pi term by multiplying one of the non repeating variables by the product of the repeating variables, each raised to an exponent that will make the combination dimensionless. Step 6. Repeat Step 5 for each of the remaining non repeating variables.

14. Step 7. Check all the resulting pi terms to make sure they are dimensionless.

15. Step 8. Express the final form as a relationship among the pi terms, and think about what it means

16. Uniqueness of Pi Terms There is not a unique set of pi terms for a given problem. Once a correct set of pi terms is obtained, any other set can be obtained by manipulation of the original set.

18. Although there is no unique set of pi terms for a given problem, the number required is fixed in accordance with the pi theorem

19. Example 1 Example 2

21. Dimensional analysis greatly facilitates the efficient handling, interpretation, and correlation of experimental data. If only one pi term is involved in a problem, it must be equal to a constant. this result is only valid for small Reynolds numbers(<<1) This equation is commonly called Stokes law

22. For problems involving only two pi terms, results of an experiment can be conveniently presented in a simple graph.

26. For problems involving more than two or three pi terms, it is often necessary to use a model to predict specific characteristics.

27. As the number of pi terms continues to increase, corresponding to an increase in the general complexity of the problem of interest, both the graphical presentation and the determination of a suitable empirical equation become intractable. It is often more feasible to use models to predict specific characteristics of the system rather than to try to develop general correlations.

29. Modeling and Similitude A model is a representation of a physical system that may be used to predict the behavior of the system in some desired respect.

30. In physical models, that is, models that resemble the prototype but are generally of a different size, may involve different fluids, and often operate under different conditions (pressures, velocities, etc.). Usually a model is smaller than the prototype. Occasionally, if the prototype is very small, it may be advantageous to have a model that is larger than the prototype so that it can be more easily studied.

31. There is, of course, an inherent danger in the use of models in that predictions can be made that are in error and the error not detected until the prototype is found not to perform as predicted. It is, therefore, imperative that the model be properly designed and tested and that the results be interpreted correctly.

32. Theory of Models This equation (1) applies to any system that is governed by the same variables irrespective of specific values for variables(such as size of components, fluid properties and so on). (1) (2) If Eq. (1) describes the behavior of a particular prototype, a similar relationship can be written for a model of this prototype

33. Variables, or pi terms, without a subscript will refer to the prototype, whereas the subscript m will be used to designate the model variables or pi terms. contains the variable that is to be predicted from observations made on the model. Therefore, if the model is designed and operated under the following conditions, (3) (4) It follows that, Prediction equation Model design condition or similarity requirements or model laws

34. Example 1 As an example of the procedure, consider the problem of determining the drag, on a thin rectangular plate placed normal to a fluid with velocity, V. We are now concerned with designing a model that could be used to predict the drag on a certain prototype The model design conditions, or similarity requirements, are therefore

35. Size of the model Geometrical scaling Thus, the required velocity for the model is obtained from the relationship Kinematic scaling From the prediction equation

36. Similarity between a model and a prototype is achieved by equating pi terms. Usually, one or more of these pi terms will involve ratios of important lengths; that is, they are purely geometrical. Thus, when we equate the pi terms involving length ratios, we are requiring that there be complete geometric similarity between the model and prototype. Sometimes complete geometric scaling may be difficult to achieve, particularly when dealing with surface roughness, since roughness is difficult to characterize and control. Any deviation from complete geometric similarity for a model must be carefully considered. Geometric similarity

37. Dynamic similarity The equality of these pi terms requires the ratio of like forces in model and prototype to be the same. Thus, when these types of pi terms are equal in model and prototype, we have dynamic similarity between model and prototype. It follows that with both geometric and dynamic similarity the streamline patterns will be the same and corresponding velocity ratios and acceleration ratios are constant throughout the flow field. Thus, kinematic similarity exists between model and prototype. Kinematic similarity

38. To have complete similarity between model and prototype, we must maintain geometric, kinematic, and dynamic similarity between the two systems. This will automatically follow if all the important variables are included in the dimensional analysis, and if all the similarity requirements based on the resulting pi terms are satisfied.

39. The ratio of a model variable to the corresponding prototype variable is called the scale for that variable. Model Scale For true models there will be only one length scale, and all lengths are fixed in accordance with this scale. The meaning of this specification is that the model is one-tenth the size of the prototype, and the tacit assumption is that all relevant lengths are scaled accordingly so the model is geometrically similar to the prototype.

40. Practical Aspects of Using Models Validation of Model Design In some situations the purpose of the model is to predict the effects of certain proposed changes in a given prototype, and in this instance some actual prototype data may be available. If the agreement is satisfactory, then the model can be changed in the desired manner, and the corresponding effect on the prototype can be predicted with increased confidence. Another useful and informative procedure is to run tests with a series of models of different sizes, where one of the models can be thought of as the prototype and the others as “models” of this prototype.

41. Distorted Models Models for which one or more similarity requirements are not satisfied are called distorted models. The classic example of a distorted model occurs in the study of open channel or free- surface flows. Typically in these problems both the Reynolds number, and the Froude number, are involved. Froude number similarity requires Reynolds number similarity requires

42. Complete similarity requires It may be quite difficult, if not impossible, to find a suitable model fluid, particularly for small length scales. For problems involving rivers, spillways, and harbors, for which the prototype fluid is water, the models are also relatively large so that the only practical model fluid is water. Generally, hydraulic models of this type are distorted and are designed on the basis of the Froude number, with the Reynolds number different in model and prototype.

43. Some Typical Model Studies Geometric and Reynolds number similarity is usually required for models involving flow through closed conduits. Flow Through Closed Conduits Common examples of this type of flow include pipe flow and flow through valves, fittings, and metering devices.

44. The first two pi terms of the right side of Eq. lead to the requirement of geometric similarity The additional similarity requirement arises from the equality of Reynolds numbers From this condition the velocity scale is established so that the actual value of the velocity scale depends on the viscosity and density scales, as well as the length scale.

45. Flow Around Immersed Bodies Geometric and Reynolds number similarity is usually required for models involving flow around bodies. Examples include flow around aircraft, automobiles, golf balls, and buildings. These types of models are usually tested in wind tunnelsaircraft Dependent pi term would usually be expressed in the form of a drag coefficient,

46. Flow with a Free Surface Froude number similarity is usually required for models involving free- surface flows. Flows in canals, rivers, spillways, and stilling basins, as well as flow around ships, are all examples of flow phenomena involving a free surface. Surface tension and viscous effects are often negligible in free-surface flows.

47. Ex 1 Ex 2 Ex 3