58 160 Intermediate Mechanics of Fluids Professor Fred Stern Typed by Stephanie Schrader Fall 2008 Chapter 5 1 Chapter 5 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 If F A1 A2 An 0 Then f 1 2 r n 0 Thereby reduces number of experiments and or simulations required to determine f vs F F functional form Ai dimensional variables j nondimensional parameters j Ai i e j consists of nondimensional groupings of Ai 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 58 160 Intermediate Mechanics of Fluids Professor Fred Stern Typed by Stephanie Schrader Fall 2008 Chapter 5 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 Theorem In a physical problem including n dimensional variables in which there are m dimensions the variables can be arranged into r n m independent nondimensional parameters r where usually m m F A1 A2 An 0 f 1 2 r 0 Ai s dimensional variables required to formulate problem i 1 n j s nondimensional parameters consisting of groupings of Ai s j 1 r F f represents functional relationships between An s and r s respectively m rank of dimensional matrix m i e number of dimensions usually 58 160 Intermediate Mechanics of Fluids Professor Fred Stern Typed by Stephanie Schrader Fall 2008 Chapter 5 3 Dimensional Analysis Methods for determining i s 1 Functional Relationship Method Identify functional relationships F Ai and f j by first determining Ai s and then evaluating j s a Inspection b Step by step Method c Exponent Method intuition text 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 j s Exponent Method for Determining j s 1 determine the n essential quantities 2 select m of the A quantities with different dimensions that contain among them the m dimensions and use them as repeating variables together with one of the other A quantities to determine each 58 160 Intermediate Mechanics of Fluids Professor Fred Stern Typed by Stephanie Schrader Fall 2008 Chapter 5 4 For example let A1 A2 and A3 contain M L and t not necessarily in each one but collectively then the j parameters are formed as follows 1 A1x1 A 2y1 A 3z1 A 4 2 A1x 2 A 2y2 A 3z 2 A 5 n m A1x n m A 2yn m A 3z n m A n Determine exponents such that i s are dimensionless 3 equations and 3 unknowns for each i In these equations the exponents are determined so that each is dimensionless This is accomplished by substituting the dimensions for each of the Ai in the equations and equating the sum of the exponents of M L and t each to zero This produces three equations in three unknowns x y t for each parameter In using the above method the designation of m m as the number of basic dimensions needed to express the n variables dimensionally is not always correct The correct value for m is the rank of the dimensional matrix i e the next smaller square subgroup with a nonzero determinant 58 160 Intermediate Mechanics of Fluids Professor Fred Stern Typed by Stephanie Schrader Fall 2008 Dimensional matrix M L t Chapter 5 5 A1 An a11 a1n a31 a3n o o o o aij exponent of M L or t in Ai n x n matrix Rank of dimensional matrix equals size of next smaller sub group with nonzero determinant Example Derivation of Kolmogorov Scales Using Dimensional Analysis Assumptions 1 Small scale motion tends to occur on a small time scale so such motion is independent of the relatively slow dynamics of the large eddies and of the mean flow 2 Kolmogorov s 1941 universal equilibrium theory The smaller eddies should be in a state where the rate of receiving energy from the larger eddies is very nearly equal to the rate at which the smallest eddies dissipate the energy to heat 3 Kolmogorov s first similarity hypothesis In every turbulent flow at sufficiently high Reynolds number the statistics of the 58 160 Intermediate Mechanics of Fluids Professor Fred Stern Typed by Stephanie Schrader Fall 2008 Chapter 5 6 small scale motions have a universal form that is uniquely determined by viscosity v and dissipation rate Facts 1 Dissipation of k to heat through the action of molecular viscosity occurs at the scale of the smallest eddies 2 Most eddies break up on a timescale of their turn over time all of EFD confirm Nomenclature u0 typical velocity associated with the largest eddies u typical velocity associated with the smallest eddies Kolmogorov scale l0 typical length scales of the largest structures typical length scales of the smallest structures Kolmogorov scale 0 typical time scales of the largest structures typical time scales of the smallest structures Kolmogorov scale Derivation 1 Based on Kolmogorov s first similarity hypothesis u f v The dimensions for v and are L2T 1 and L2T 3 respectively Herein L and T are the units for length and time respectively There are to within multiplicative constants unique length velocity and time scales that can be formed which are the Kolmogorov scales Herein the exponential method is used 58 160 Intermediate Mechanics of Fluids Professor Fred Stern Typed by Stephanie Schrader Fall 2008 Chapter 5 7 F N u vN N 0 N L2 L2 L N L T T T3 T n 5 1 use v and as repeating variables m 2 r n m 3 1 v x1 y1 L2T 1 L2T 3 L x1 y1 L 2 x1 2 y1 1 0 T x1 3 y1 0 2 3 x1 3 4 and y1 1 4 v x y 1 1 3 4 v 1 4 v3 4 14 2 v x2 y2 u L2T 1 L T LT x2 3 y2 2 1 L 2 x2 2 y2 1 0 T x2 3 y2 1 0 5 6 x2 y2 1 4 u v x2 y2 1 4 1 4 v v 14 7 3 v x3 y3 L2T 1 x3 L2T 3 y3 T 8 58 160 Intermediate Mechanics of Fluids Professor Fred Stern Typed by Stephanie Schrader Fall 2008 Chapter 5 8 L 2 x3 2 y3 0 T x3 3 y3 1 0 9 x3 1 2 and y3 1 2 v x y 3 1 2 v 3 1 2 v 12 10 v3 14 u v 14 v 12 11 From Eqn 11 the dissipation rate can be expressed by v u v 2 2 Showing that u 1 12 13 which provides a consistent characterization of the velocity gradients of the dissipative eddies Also from Eqn …
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