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Unformatted text preview: Question 1: X51 X52 X6A X48 X61 X32 X31 X33 X67 X4C When a jet of ﬂuid impinges on a horizontal plate, the ﬂuid ﬂows radially outward away from the jet in a thin film. At a distance R j from the jet, the thickness of the film suddenly increases. This phenomenon is known as a hydraulic jump. In general, the ﬂow in the jump region is turbulent. 1 Consider a jet with volume ﬂux, Q , and radius, a . The ﬂuid in the jet can be approximated as inviscid with density, ρ . The acceleration of gravity is − g . You may assume the ﬂow is axially symmetric. You may also assume a H R j . The hydraulic jump can be divided into three regions as indicated in the figure: (1) the upstream region, a < r < R j , (2) the jump region r ∼ R j and (3) the downstream region r > R j . Follow the steps below to find an expression for the steady state jump radius, R j . a.) Using dimensional analysis, find a complete set of independent Pi groups for this system. Solution: The jump radius could depend on: R j = Φ( ρ,Q,a,H,g ) However, the only parameter in this list that includes mass is ρ so there is no way to eliminate [ M ] if ρ in included in a Pi group. Hence R j = Φ( Q,a,H,g ) Whether we include H as a parameter depends on the far-field boundary con- dition. In many hydraulic jump experiments, H can be tuned independently 1 This turbulence may be suppressed in suﬃciently viscous ﬂuids but here we will only consider the turbulent case. 1 Photo removed for copyright reasons. by placing a wall of height H at a radius much larger than the jump radius. However H may also be determined by a balance of gravity and surface tension. Since the far-field conditions were not specified, either answer is acceptable. L may also be included as a possible paramter. µ should not be included as the ﬂow is modeled as inviscid. The number of Pi groups = n − k (using the list above) where n = 5 and k = 2 ([ L ] and [ T ]). Thus H Π 3 = a 5 g Π 1 = R j , Π 2 = , a a Q 2 Note that Π 3 = a 5 g/Q 2 ∼ ga/v 2 = 1 /F r a where F r a is the Froude number defined using a as a characteristic length scale. b.) In the downstream region (3), the height of the free surface is a known constant, H (i.e., the height of the free surface is NOT a function of r ). Find an expression for the average velocity in the downstream region. Solution: By conservation of mass: Q = 2 πrHv r ( r ) ⇒ v r ( r ) = Q 2 πHr c.) The upstream region (1) consists of a jet impinging on a horizontal plate. Derive an expression for the height of the free surface in this region, h ( r ), and the average radial velocity. Recall that you may assume a R j . Solution: X31 X32 By conservation of mass: Q = v 1 πa 2 = v 2 2 πrh ( r ) Following a streamline from point 1 to point 2 along the free surface, we can apply Bernoulli (note that p = p a everywhere along the streamline)....
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This note was uploaded on 02/27/2012 for the course MECHANICAL 2.25 taught by Professor Garethmckinley during the Fall '05 term at MIT.
- Fall '05