Chapter7 - 57:020 Mechanics of Fluids and Transport...

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Unformatted text preview: 57:020 Mechanics of Fluids and Transport Processes Chapter 7 Professor Fred Stern Fall 2006 1 Chapter 7 Dimensional Analysis and Modeling The Need for Dimensional Analysis Dimensional analysis is a process of formulating fluid mechanics problems in terms of nondimensional variables and parameters. 1. Reduction in Variables: F = functional form If F(A 1 , A 2 , …, A n ) = 0, A i = dimensional variables Then f( Π 1 , Π 2 , … Π r < n ) = 0 Π j = nondimensional p a r a m e t e r s Thereby reduces number of = Π j (A i ) experiments and/or simulations i.e., Π j consists of required to determine f vs. F nondimensional groupings of A i ’s 2. Helps in understanding physics 3. Useful in data analysis and modeling 4. Fundamental to concept of similarity and model testing Enables scaling for different physical dimensions and fluid properties 57:020 Mechanics of Fluids and Transport Processes Chapter 7 Professor Fred Stern Fall 2006 2 Dimensions and Equations Basic dimensions: F, L, and t or M, L, and t F and M related by F = Ma = MLT-2 Buckingham Π Theorem In a physical problem including n dimensional variables in which there are m dimensions, the variables can be arranged into r = n – m ˆ independent nondimensional parameters Π r (where usually m ˆ = m). F(A 1 , A 2 , …, A n ) = 0 f( Π 1 , Π 2 , … Π r ) = 0 A i ’s = dimensional variables required to formulate problem (i = 1, n) Π j ’s = nondimensional parameters consisting of groupings of A i ’s (j = 1, r) F, f represents functional relationships between A n ’s and Π r ’s, respectively m ˆ = rank of dimensional matrix = m (i.e., number of dimensions) usually 57:020 Mechanics of Fluids and Transport Processes Chapter 7 Professor Fred Stern Fall 2006 3 Dimensional Analysis Methods for determining Π i ’s 1. Functional Relationship Method Identify functional relationships F(A i ) and f( Π j )by first determining A i ’s and then evaluating Π j ’s a. Inspection intuition b. Step-by-step Method text c. Exponent Method class 2. Nondimensionalize governing differential equations and initial and boundary conditions Select appropriate quantities for nondimensionalizing the GDE, IC, and BC e.g. for M, L, and t Put GDE, IC, and BC in nondimensional form Identify Π j ’s Exponent Method for Determining Π j ’s 1) determine the n essential quantities 2) select m ˆ of the A quantities, with different dimensions, that contain among them the m ˆ dimensions, and use them as repeating variables together with one of the other A quantities to determine each Π . 57:020 Mechanics of Fluids and Transport Processes Chapter 7 Professor Fred Stern Fall 2006 4 For example let A 1 , A 2 , and A 3 contain M, L, and t (not necessarily in each one, but collectively); then the Π j parameters are formed as follows: n z 3 y 2 x 1 m n 5 z 3 y 2 x 1 2 4 z 3 y 2 x 1 1 A A A A A A A A A A A A m n m n m n 2 2 2 1 1 1 − − − = Π = Π = Π − In these equations the exponents are determined so that...
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Chapter7 - 57:020 Mechanics of Fluids and Transport...

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