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UI ME 5160 - Intermediate Mechanics of Fluids

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58:160 Intermediate Mechanics of Fluids Chapter 5 Professor Fred Stern Typed by Stephanie Schrader Fall 2008 1Chapter 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: F = functional form If F(A1, A2, …, An) = 0, Ai = dimensional variables Then f(Π1, Π2, … Πr < n) = 0 Πj = nondimensional parameters Thereby reduces number of = Πj (Ai) experiments and/or simulations i.e., Πj consists of required to determine f vs. F 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 properties58:160 Intermediate Mechanics of Fluids Chapter 5 Professor Fred Stern Typed by Stephanie Schrader Fall 2008 2Dimensions 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 – mˆ independent nondimensional parameters Πr (where usually mˆ = m). F(A1, A2, …, An) = 0 f(Π1, Π2, … Πr) = 0 Ai’s = dimensional variables required to formulate problem (i = 1, n) Πj’s = nondimensional parameters consisting of groupings of Ai’s (j = 1, r) F, f represents functional relationships between An’s and Πr’s, respectively mˆ = rank of dimensional matrix = m (i.e., number of dimensions) usually58:160 Intermediate Mechanics of Fluids Chapter 5 Professor Fred Stern Typed by Stephanie Schrader Fall 2008 3Dimensional Analysis Methods for determining Πi’s 1. Functional Relationship Method Identify functional relationships F(Ai) and f(Πj)by first determining Ai’s and then evaluating Πj’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 Πj’s Exponent Method for Determining Πj’s 1) determine the n essential quantities 2) select mˆ of the A quantities, with different dimensions, that contain among them the mˆ dimensions, and use them as repeating variables together with one of the other A quantities to determine each Π.58:160 Intermediate Mechanics of Fluids Chapter 5 Professor Fred Stern Typed by Stephanie Schrader Fall 2008 4For example let A1, A2, and A3 contain M, L, and t (not necessarily in each one, but collectively); then the Πj parameters are formed as follows: nz3y2x1mn5z3y2x124z3y2x11AAAAAAAAAAAAmnmnmn222111−−−=Π=Π=Π− In these equations the exponents are determined so that each Π 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 Π parameter. In using the above method, the designation of mˆ = m as the number of basic dimensions needed to express the n variables dimensionally is not always correct. The correct value for mˆ is the rank of the dimensional matrix, i.e., the next smaller square subgroup with a nonzero determinant. Determine exponents such that Πi’s are dimensionless 3 equations and 3 unknowns for each Πi58:160 Intermediate Mechanics of Fluids Chapter 5 Professor Fred Stern Typed by Stephanie Schrader Fall 2008 5 Dimensional matrix = A1 ……… An M a11 ……… a1n L t a31 ……… a3n o ……… o : : : : : : o ……… o 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 aij = exponent of M, L, or t in Ai58:160 Intermediate Mechanics of Fluids Chapter 5 Professor Fred Stern Typed by Stephanie Schrader Fall 2008 6small-scale motions have a universal form that is uniquely determined by viscosity v and dissipation rate ε. 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 0u ---- typical velocity associated with the largest eddies uη---- typical velocity associated with the smallest eddies (Kolmogorov scale) 0l ---- typical length scales of the largest structures η ---- typical length scales of the smallest structures (Kolmogorov scale) 0τ ---- typical time scales of the largest structures ητ---- typical time scales of the smallest structures (Kolmogorov scale) Derivation 1: Based on Kolmogorov’s first similarity hypothesis, ()(),, ,ufvηηητε=. The dimensions for v and ε are 21LT−and 23LT−, 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 Chapter 5 Professor Fred Stern Typed by Stephanie Schrader Fall 2008 7 NNNNN223,,,, 0 5LLLLTTTTFu v nηηητ ε⎛⎞⎜⎟==⎜⎟⎜⎟⎜⎟⎝⎠ (1) use v and ε as repeating variables, m=2⇒ r=n-m=3 ()()1111121 23xyxyvLT LT Lεη−−∏== (2) 1111221030LxyTxy++=−− = (3) x1=-3/4 and y1=1/4


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