Ch07_1_DA - MAE130B Ch 7(1 Dimensional Analysis Changzheng...

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MAE130B – Ch 7 (1) Dimensional Analysis Changzheng Huang
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Ch 7 (1) Changzheng Huang 2 1. Dimensional Analysis 1.1 WHAT is dimensional analysis? 1.2 WHY need dimensional analysis? 1.3 HOW to do dimensional analysis?
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Ch 7 (1) Changzheng Huang 3 1.1 WHAT is dimensional analysis? Dimensions and units Physical/Fluid Quantities Dimension Unit Length m or ft Time s Velocity m/s or ft/s Example: The wind is blowing at a speed of 50 miles/hour .
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Ch 7 (1) Changzheng Huang 4 1.1 WHAT is dimensional analysis? Basic dimensions FLT system: Force F Length L Time T Temperature Θ MLT system: Mass M Length L Time T Temperature Θ These primary quantities are called basic dimensions. F=M × a Newton’s Law
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Ch 7 (1) Changzheng Huang 5 1.1 WHAT is dimensional analysis? Dimensions of some common quantities (Table 1.1, page 4) FLT system MLT system Force F MLT -2 Mass FL -1 T 2 M Length L L Time T T Velocity LT -1 LT -1 Pressure FL -2 ML -1 T -2 Density FL -4 T 2 ML -3 Viscosity (dynamic) FL -2 TM L -1 T -1 Modulus of elasticity FL -2 ML -1 T -2 How about acceleration in FLT and MLT ?
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Ch 7 (1) Changzheng Huang 6 1.1 WHAT is dimensional analysis? Dimensions and physical laws Any equation describing a physical law must be dimensionally homogeneous, • Dimensions on left side must be same as those on right side. • All additive separate terms must have the same dimensions. () ( ) 22 11 2 1 42 1 2 2 33 3 3 uu u p u u uv tx y x x y LT LT FL LT FL T FL T LT FL T TL L L FL FL FL FL ρρ ρ µ −− ∂∂ ++ = + + ± ± An apple-to-apple comparison.
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Ch 7 (1) Changzheng Huang 7 1.1 WHAT is dimensional analysis? Dimensional analysis Dimensional analysis is an analysis method that is based on the analysis of physical dimensions. It can help us reduce the complexity and gain insights in the study of flow phenomena.
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Ch 7 (1) Changzheng Huang 8 1. Dimensional Analysis 1.1 WHAT is dimensional analysis? 1.2 WHY need dimensional analysis? 1.3 HOW to do dimensional analysis?
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Ch 7 (1) Changzheng Huang 9 1.2 WHY need dimensional analysis? An example , ρµ V D l 12 l p p p l ∆= ( ) ,,, l pf DV = Task: how to develop an experimental plan that will help to determine the functional relationship?
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Ch 7 (1) Changzheng Huang 10 1.2 WHY need dimensional analysis? Plan I (from a seemingly sound approach) ( ) ,,, l pf DV ρµ = VD ρ µ l p l p l p l p , , constant D ρµ− , , constant V , , constant , , constant
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Ch 7 (1) Changzheng Huang 11 1.2 WHY need dimensional analysis? Plan II (from the dimensional analysis approach) () ,,, l pf DV ρµ ∆= VD ρ µ 2 l Dp V 2 l VD V φ ⎛⎞ = ⎜⎟ ⎝⎠ ( ) ( ) ( ) 3 00 0 2 2 42 1 1 2 l LF L FLT V FL T LT FL T LT L VD FL T −− = == ± Advantages: • reduce number of variables. • generalize flow phenomenon.
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Ch07_1_DA - MAE130B Ch 7(1 Dimensional Analysis Changzheng...

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