ME 3560 Fluid Mechanics

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Ethelbert Allison
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1 ME 3560 Fluid Mechanics 1

2 7.1 Dimensional Analysis Many problems of interest in fluid mechanics cannot be solved using the integral and/or differential equations. Examples of problems that are studied in the laboratory with the use of models are: Wind motions around a football stadium. The flow of water through a large hydro-turbine. The airflow around the deflector on a semi-truck. The wave motion around a pier or a ship. Airflow around aircraft. 2

3 A laboratory study with the use of models can be very expensive. To minimize the cost, dimensionless parameters are used. Such parameters are also used in numerical studies for the same reason. Dimensionless parameters are obtained using a method called dimensional analysis. Dimensional Analysis is based on the idea of dimensional homogeneity: all terms in an equation must have the same dimensions. By using this idea, it is possible to minimize the number of parameters needed in an experimental or analytical analysis. 3

4 Any equation can be expressed in terms of dimensionless parameters simply by dividing each term by one of the other terms. 2 2 p1 V1 p2 V2 Consider g z1 g z Divide both sides by gz 2 to obtain: p2 V 2 p1 V 1 z 1 1 gz2 2gz 2 gz1 2gz 1 z The dimensionless parameters, V 2 / gz and p / gz canbeusedtopredict the performance of a prototype with a model tested in the laboratory. 1 2 Similitude is the study that allows the prediction of the quantities to be expected on a prototype from the measurements on a model. 4

5 Thus, Dimensional Analysis and Similitude seek to approach the following problems: Express a given dimensional, functional relation in a Non dimensional form. Use Non dimensional parameters in similitude testing of models to use the results obtained to predict performance of prototypes. Non dimensionalize equations (algebraic or differential) to determine relevant non dimensional parameters. 5

6 7.2 Buckingham Pi Theorem The number of dimensionless products required to replace the original list of variables describing a physical phenomenon is established by Buckingham Pi Theorem, which is the basic theorem of dimensional analysis and states that: If an equation involving k variables is dimensionally homogeneous, it can be reduced to a relationship among k r independent dimensionless products, where r is the minimum number of reference dimensions required to describe the variables. The dimensionless products are frequently referred to as pi terms 6

7 The Pi theorem is based on the idea of dimensional homogeneity. It is assumed that for any physically meaningful equation involving k variables, such as: u1 f ( u2, u3,..., uk ) The dimensions of the variable on the L.H.S. of the equal sign must be equal to the dimensions of any term that stands by itself on the R.H.S. of the equation. Thus, it is possible to rearrange the equation into a set of dimensionless products (pi terms) so that 1 ( 2, 3,..., k r ) The required number of pi terms is fewer than the number of original variables by r. r is the minimum number of reference dimensions required to describe the original list of variables. 7

8 Usually the reference dimensions required to describe the variables will be the basic dimensions M, L, andt or F, L, andt. In some instances perhaps only two dimensions, such as L and T, are required, or maybe just one, such as L. In a few rare cases the variables may be described by some combination of basic dimensions, such as M/T 2 and L, and in this case r would be equal to two rather than three. 8

9 7.3 Determination of the Pi Terms 1. List all the variables involved in the problem. 2. Express each of the variables in terms of basic dimensions. 3. Determine the required number of Pi terms. 4. Select a number of repeating variable, where the number required is equal to the number of reference dimensions. 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. 6. Repeat the previous step for the remaining non repeating variables. 7. Check all resulting Pi terms to make sure they are dimensionless and independent. 8. Express the final form as a relationship between Pi terms. 9

10 Factors to Consider when Selecting the Variables Involved in a Phenomenon: Geometry. The geometric characteristics can usually be described by a series of lengths and angles. In most problems the geometry of the system plays an important role, and a sufficient number of geometric variables must be included to describe the system. Material Properties. Since the response of a system to applied external effects such as forces, pressures, and changes in temperature is dependent on the nature of the materials involved in the system, the material properties that relate the external effects and the responses must be included as variables. External Effects. This terminology is used to denote any variable that produces, or tends to produce, a change in the system. For example, forces applied to a system, pressures, velocities, or gravity. 10

11 When selecting the variables involved in a problem it is also needed to: Keep the number of variables to a minimum, Make sure that all variables selected are independent. Clearly define the problem. What is the main variable of interest (the dependent variable)? Consider the basic laws that govern the phenomenon. Start the variable selection process by grouping the variables into three broad classes: geometry, material properties, and external effects. Consider other variables that may not fall into one of the above categories. For example, time will be an important variable if any of the variables are time dependent. Be sure to include all quantities that enter the problem even though some of them may be held constant (e.g., the acceleration of gravity, g). 11

12 Problem 7.12: At a sudden contraction in a pipe the diameter changes from D 1 to D 2. The pressure drop, Δp, which develops across the contraction is a function of D 1 and D 2, as well as the velocity, V, inthe larger pipe, and the fluid density,, and viscosity, μ. UseD 1, V, andμ as repeating variables to determine a suitable set of dimensionless parameters. Why would it be incorrect to include the velocity in the smaller pipe as an additional variable? Problem 7.17: A thin elastic wire is placed between rigid supports. A fluid flows past the wire, and it is desired to study the static deflection, δ, at the center of the wire due to the fluid drag. Assume that: f ( l, d,,, E, V ) where l is the wire length, d the wire diameter, the fluid density, μ the fluid viscosity, V the fluid velocity, and E the modulus of elasticity of the wire material. Develop a suitable set of pi terms for this problem. 12

13 7.6 Common Dimensionless Groups in Fluid Mechanics 13

7.8 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.

Mathematical or computer models may also conform to this definition, but our interest will be in physical models: models that resemble the prototype but are generally of a

14 7.8 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. The physical system for which the predictions are to be made is called the prototype. Mathematical or computer models may also conform to this definition, but our interest will be in physical models: 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. 14

15 7.8.1 Theory of Models The process to determine the performance of a prototype based on the study of a model can be summarized as follows: 1. Establish the dependent variable and the variables that affect it. u1 f ( u2, u3,..., uk ) 2. Determine the Pi terms for the problem: 1 ( 2, 3,..., k r ) 3. Prepare a model and experimental conditions such that the Pi terms in the model match those in the prototype: 2 m 2 p; 3m 3 p;... ( k r) m ( k r) p Therefore: 1p 1m 15

16 By matching the Pi terms between model and prototype, geometric similitude, kinematic similitude and dynamic similitude are matched. Lp Geometric Similitude: Scaleratio Typically 1 Lm Kinematic Similitude: It is related to velocity, angular velocity, acceleration, etc. between the model and the prototype Dynamic Similitude: It is related to fluid properties, forces, moments, pressures, etc. 16

BUCKINGHAM PI THEOREM

BUCKINGHAM PI THEOREM Dimensional Analysis It is used to determine the equation is right or wrong. The calculation is depends on the unit or dimensional conditions of the equations. For example; F=ma F=MLT