HW_SET_1.soln - Chapter 1 Introduction 9 1.14 The volume...

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Chapter 1 Introduction 9 1.14 The volume flow Q over a dam is proportional to dam width B and also varies with gravity g and excess water height H upstream, as shown in Fig. P1.14. What is the only possible dimensionally homo- geneous relation for this flow rate? Solution: So far we know that Q B fcn(H,g). Write this in dimensional form: Fig. P1.14 3 L {Q} {B}{f(H,g)} {L}{f(H,g)},    2 T L or: {f(H,g)} T  So the function fcn(H,g) must provide dimensions of {L 2 /T}, but only g contains time . Therefore g must enter in the form g 1/2 to accomplish this. The relation is now Q Bg 1/2 fcn(H), or: {L 3 /T} {L}{L 1/2 /T}{fcn(H)}, or: {fcn(H)} { L 3/2 }In order for fcn(H) to provide dimensions of {L 3/2 }, the function must be a 3/2 power. Thus the final desired homogeneous relation for dam flow is: Q C B g 1/2 H 3/2 , where C is a dimensionless constant Ans. P1.15 Mott [49] recommends the following formula for the friction head loss h f , in ft, for flow through a pipe of length L o and diameter D (both in ft): 852 . 1 63 . 0 ) ( 551 . 0 D AC Q L h h o f where Q is the volume flow rate in ft 3 /s, A is the pipe cross-section area in ft 2 , and C h is a dimensionless coefficient whose value is approximately 100. Determine the dimensions of the constant 0.551.
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10 Solutions Manual Fluid Mechanics, Sixth Edition Solution : Write out the dimensions of each of the terms in the formula: Use these dimensions in the equation to determine {0.551}. Since h f and L o have the same dimensions { L }, it follows that the quantity in parentheses must be dimensionless: The constant has dimensions; therefore beware . The formula is valid only for water flow at high ( turbulent ) velocities. The density and viscosity of water are hidden in the constant 0.551, and the wall roughness is hidden (approximately) in the numerical value of C h . 1.16 Test the dimensional homogeneity of the boundary-layer x -momentum equation: x uu p uv g xy x y    Solution: This equation, like all theoretical partial differential equations in mechanics, is dimensionally homogeneous. Test each term in sequence:     3 M L L / T p M / L T ; T L x L L 22 MM LT 2 2 x 32 ML M /LT {g} ; xL  All terms have dimension {ML –2 T –2 }. This equation may use any consistent units.
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This note was uploaded on 10/12/2011 for the course CIVIL 2000 taught by Professor Dong during the Spring '11 term at Miami University.

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HW_SET_1.soln - Chapter 1 Introduction 9 1.14 The volume...

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