DIMENSIONAL ANALYSIS, SIMILARITY AND MODELING

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CHAPTER 7 OUTLINE

Dimensional analysis Buckingham PI Theorem

Similarity Modeling and Prototype

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OBJECTIVES

‣ Develop a better understanding of dimensions, units,

and dimensional homogeneity of equations ‣ Understand the numerous benefits of dimensional

analysis ‣ Know how to use the method of repeating variables to

identify non-dimensional parameters ‣ Understand the concept of dynamic similarity and how

to apply it to experimental modeling

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DIMENSIONS AND UNITSDimension: A measure of a physical quantity (without numerical values). Unit: A way to assign a number to that dimension. There are seven primary dimensions (also called fundamental or basic dimensions): mass, length, time, temperature, electric current, amount of light, and amount of matter. All non-primary dimensions can be formed by some combination of the seven primary dimensions

A dimension is a measure of a physical quantity without numerical values, while a unit is a way to assign a number to the dimension. For example, length is a dimension, but centimeter is a unit.

DIMENSIONAL ANALYSIS Dimensional analysis is a method for reducing the number and complexity of experimental variables that affect a given physical phenomenon, by using a sort of compacting technique.

Primary purposes of dimensional analysis ‣ To generate non-dimensional parameters that help in the

design of experiments (physical and/or numerical) and in reporting of results

‣ To obtain scaling laws so that prototype performance can be predicted from model performance.

‣ To predict trends in the relationship between parameters.

The benefits of dimensional analysis

‣ Saving in time and money ‣ Helps planning for coming experiment or theory/simulation

(before we spend money on computer analysis/simulation) ‣ Provides scaling laws that can convert data from a cheap, small

model to design information for an expensive, large prototype. ‣ E.g : The force, F on a particular body shape immersed in a fluid

stream is depend on the body length, L, fluid velocity, V, fluid density, and fluid viscosity, µ or F = f(L, V, , µ ).

‣ Because of the geometry and flow condition are so complicated, the theory fail to yield the solution for F. Therefore, the function of f(L, V, , µ ) must be find experimentally (or numerically). If we want to predict the effect of L to the F, we have to run the experiment for 10 length L. For each L we need 10 values of V, 10 value of and 10 value µ. Or in other words we have to run about 10 000 experiments which involved a very high cost. If each experiment is cost about RM…, totally we have to spend about RM….

‣ When using the dimensional analysis, we could reduce the variable involved becomes, F/(�V2L2) = g (�VL/µ). The experiment need to run only for 10 value of Re not for each single variable of L, V, or µ.

The Buckingham PI Theorem

There are some method can be used in predicting the nondimensional parameters/group - Rayleigh, Buckingham, Step–by-step (Ipsen).

The most popular method – the Theorem of Buckingham (Edgar Buckingham 1867-1940)

Based on two theorem of Buckingham : If a problem involving m variables, it can be reduced to a

relationship among independent dimensionless products, where n is the minimum number of reference dimensions required to describe the variables.

Each of group is consist of n repeating variables and one non-repeating variable.

7 steps in doing 𝝅 Buckingham analysis

1. List and count m variables involved. If any important variables are missing, dimensional analysis will fail.

2. List the dimensions of each variable and count n where n = number of primary dimension involved (according to MLT OR FLT system)

3. Determine the group exist where = m – n 4. Select n repeating variables (based on the guideline in selecting the

repeating variables) A. Geometry similarity B. Kinematic similarity C. Dynamic similarity

5. Form the group of ’s with combining the repeating variables with one of the non-repeating variable.

6. Check all the resulting terms to make sure they are dimensionless. 7. Express the final form as a relationship among the terms as : 1 = Φ(2, 𝝅3,…, 𝝅m-n)

TEXT

EXAMPLE 7.1

Force F induced on a propeller blade is a function of diameter of propeller D, velocity of fluid u, density of fluid , dynamic viscosity and rotation of propeller N. Determine the non-dimensional groups.

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MODELING AND PROTOTYPE SIMILARITYIn most experiments, to save time and money, tests are performed on a

geometrically scaled model, rather than on the full-scale prototype.

In such cases, care must be taken to properly scale the results. We introduce here a powerful technique called dimensional analysis.

The three primary purposes of dimensional analysis are • To generate nondimensional parameters that help in the design of experiments

(physical and/or numerical) and in the reporting of experimental results • To obtain scaling laws so that prototype performance can be predicted from

model performance

• To (sometimes) predict trends in the relationship between parameters The principle of similarity Three necessary conditions for complete similarity between a model and a

prototype.

(1) Geometric similarity—the model must be the same shape as the prototype, but may be scaled by some constant scale factor.

(2) Kinematic similarity—the velocity at any point in the model flow must be proportional (by a constant scale factor) to the velocity at the corresponding point in the prototype flow.

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MODELING AND PROTOTYPE SIMILARITY (CONT.)

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(3) dynamic similarity—When all forces in the model flow scale by a constant factor to corresponding forces in the prototype flow (force-scale equivalence).

In a general flow field, complete similarity between a model and prototype is achieved only when there is geometric, kinematic, and dynamic similarity.

n Kinematic similarity is achieved when, at all locations, the speed in the model flow is proportional to that at corresponding locations in the prototype flow, and points in the same direction.

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MODELING AND PROTOTYPE SIMILARITY (CONT.)

Geometric Similarity -  the model must have the same shape as the prototype. -  each length dimension must be scaled by the same factor.

E.g : For above one-tenth-scale model of prototype wing -  the model length are 1/10 as large , but the angle of attack is

remain the same : 10° not 1°. -  Model nose is 1/10 as large -  Model surface roughness is 1/10 as large -  If the prototype is constructed with protruding fasteners, the

model also should have homologous protruding fasterners 1/10 as large.

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MODELING AND PROTOTYPE SIMILARITY (CONT.)

Kinematic Similarity Velocity at any point in the model must be proportional (velocity scale ratio must be the same for both scale model and prototype).

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MODELING AND PROTOTYPE SIMILARITY (CONT.)Dynamic Similarity All forces in the model flow scale by a constant factor to corresponding forces in the prototype flow.

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MODELING AND PROTOTYPE SIMILARITY (CONT.)

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n We let uppercase Greek letter Pi (Π) denote a nondimensional parameter. n In a general dimensional analysis problem, there is one Π that we call the dependent Π, giving it the notation Π1.

n The parameter Π1 is in general a function of several other Π�s, which we call independent Π�s.

n To ensure complete similarity, the model and prototype must be geometrically similar, and all independent groups must match between model and prototype.

n To achieve similarity

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Geometric similarity between a prototype car of length Lp and a model car of length Lm.

The Reynolds number Re is formed by the ratio of density, characteristic speed, and characteristic length to viscosity. Alternatively, it is the ratio of characteristic speed and length to kinematic viscosity, defined as ν =µ/ρ.

The Reynolds number is the most well known and useful dimensionless parameter in all of fluid mechanics.

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n A drag balance is a device used n in a wind tunnel to measure the n aerodynamic drag of a body. When testing automobile models, a moving belt is often added to the floor of the wind tunnel to simulate the moving ground (from the car�s frame of reference).

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n A drag balance is a device used n in a wind tunnel to measure the n aerodynamic drag of a body. When testing automobile models, a moving belt is often added to the floor of the wind tunnel to simulate the moving ground (from the car�s frame of reference).

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END OF CHAPTER 7 THANK YOU

Dr. Shahrul Azmir Osman

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