Convection in Three-Dimensional Separated and Attached Flow

Convection in Three-Dimensional Separated and Attached Flow B. F. Armaly Convection Heat Transfer Laboratory Department of Mechanical and Aerospace Engineering, and Engineering Mechanics University of Missouri Rolla

Contents Introduction Results and Discussions Effects of Reynolds number Effects of step height Effects of aspect ratio Effects of buoyancy force Conclusions

Introduction Separated and reattached flow occurs in many industrial engineering applications; The heat transfer rate increases greatly and reaches its maximum value in the reattachment region; A great deal of mixing of high and low energy fluid occurs in the separated and reattached flow regions, thus impacting significantly the heat transfer performance of these devices.

Flow around the car, and major locations of flow separation

Flow separation around the airfoil

Electronics cooling

Separated flow has been studied extensively during the past decades, and flow adjacent to backward-facing step received most of the attention. It has been used as benchmark problem; Most of the published results are limited to twodimensional flow; But, flow is three-dimensional in most of the actual systems;

h u H S -D single-sided backward-facing step I: Corner eddy IV: Reattached zone II: Reverse flow zone V: Redeveloping near-wall flow III: Separated shear layer VI: Relaxing outer shear layer

5 Thousands of Transistors 4 3 Intel CPUs 886 886 P7 (Merced) P6 (Pentium Pro) P5 (Pentium) 8486 8386 44 97 975 98 985 99 995 Year Moore's Law for Intel CPUs The data used to construct this graph have been adapted from the Microprocessor Report 9(6), May 995

Flow patterns Flow T = o C S h z y x (,,) Side wall Y Symmetry center plane Heated bottom wall H L Z X q w q w = 5W/m Step height (S) is mm; Duct s height upstream of the step (h) is mm; Duct s height downstream of the step (H) is mm; Aspect ratio (AR = W/S) is 8; Expansion ratio (ER = H/h) is.

Governing equations Continuity equation: ρ ρ ρ x u y v z w ( ) ( ) ( ) ; + + = Momentum equations: ρ ρ ρ µ x u y uv z uw p x u x u y u z ( ) ( ) ( ) + + = + + + ( ) + ρ β g T T ρ ρ ρ µ x uv y v z vw p y v x v y v z ( ) ( ) ( ) + + = + + + ρ ρ ρ µ x uw y vw z w p z w x w y w z ( ) ( ) ( ) + + = + + + u x CT v y CT w z CT k T x T y T z P P P ρ ρ ρ ( ) ( ) ( ) + + = + + Energy equation:

6 Re = 648 8 Re = 397 4 Armaly et al. (983) Present results..4.6.8 Comparison of the predicted results with the measurements AR = 36

Flow Direction Re = 4 3 Step y/s Z Y 5 X x u / S General flow patterns adjacent to backward-facing step A downwash flow with a vortex flow inside primary recirculation region; A jet-like flow that moves toward the stepped wall; 3 A reverse flow region adjacent to the sidewall. 5.5

Flow Direction Re = 4 =.5 Z Y X 5 x u /S 5.5 y/s Streamlines starting from the spanwise plane near the sidewall ( =.5), as a sample of the planes between < <.3

Flow Direction Re = 4 =.75 Z Y X 5 x u /S 5.5 y/s Streamlines starting from the spanwise plane of =.75, as a sample of the planes between.3 < <.5

y/s Flow Direction Re = 4 =.6 Y Z X (Viewed from the side) Flow Direction Re = 4 =.6 Step Y Z Y Z X X 5 x u /S 5 Flow Direction Re = 4 =.6.5 (Viewed from the top) Step 5 5 The incoming fluid at this plane does not reattach directly to the stepped wall. y/s Step 5 5.5 The effects of spanwise flow can be seen in planes that are below and above the step. Streamlines starting from the spanwise plane of ( =.6), as a sample of the planes between.5 < <

Streamlines starting from the spanwise plane of ( =.6), as a sample of the planes between.5 < <

Flow D irect io n Re = 4 Re = 4 y/s x u /S Y.5 5 Re = 4 Flow Direction y/s (Viewed from the side) 5 Y Step Z X.5 5 X Z Y Z X L S tep z/ y/s (Viewed from the exit) 5 Recirculation region adjacent to the sidewall. A jet-like flow from this recirculation region impinges on the stepped wall. The jet-like flow and the recirculation region that develops adjacent to the sidewall

Recirculation region adjacent to the sidewall and the jet-like flow starting from it

5 5 Reference velocity vector:. m/s Re = 4 u y w y v = = = x b vm/s ( ).8E-5 9.6E-6.E-6-7.E-6 -.5E-5 -.4E-5-3.E-5-4.E-5-4.9E-5-5.7E-5.5.5.75 Velocity field on a y-plane adjacent to the stepped wall (y/s =.)

Cf Flow Direction Experimental.33..56.78.56.67...44.5 Re = 343.. Z Y 5 5.5 X 5 x u /S 5 Friction coefficient distribution

Flow Direction Re = 343.333.556.778.333.889..778..333.556.5 Nu 5 x u /S 5 5 5 Experimental.5 Z Y X Nusselt number distribution

Lighting system Electric heater TLC surface Tranverse Cooling water Digital Camera Base support Schematic of TLC calibration facility

5 Temperature( C) 36.93 36.74 35.86 35.45 34.8 34.49 33.93 33.6 3.7 3.48 3.43 5 4 6 8 TLC image and temperature distribution Re = 343 q w = 6.3w/m

Schematic of wind tunnel and test section

Laser Doppler velocimeter (LDV)

5 x S 5 5 Locations of detachment and reattachment of the flow at the center of the test section (with changing Reynolds number)

Effects of Reynolds number 8 6 4 3 4 5 6 7 8 Re 3 Effects of Reynolds number on the reattachment length at the center

5 * Re = 99. Re = 4.4 x Re = 47.3 Re = 346.7 Re = 36.7 x x x x x xxx x x x x x x x xxx x x x x x x x x x x x x x x x xxx xxx x x 5 * ** ****** * * * * * * ** * * ** * * * * * * * * * * * ******* **.5.5 Spanwise distribution of the reattachment length

.5 Comparisons of predictions and measurements Re=98.5 } Re=343 Exp. Re =5 Simulation y/s 4 8 Re = 343 Re = 5}Experimental Numerical 5 5.4.8.4.8.4.8.4.8.4.8.4.8.4.8.4.8.4.8 u(m/s) y/s.5.5 Re = 55.5 3 5 7 5 5

.5 = =5 y/s.5.5.75.5.75.975 =7 y/s.5 =3..4..4 u(m/s) u(m/s) Profiles of u-velocity component at several x-planes

Effects of step height 4 ER =.67 ER =. ER =.5 Experimental [3] 8 6 Re = const (343) 4 Locations of the "jet-like" flow.5 Effects of step height on distribution of the reattachment length

y/s 5 5.5 Flow Direction x r /S Y X Z ER =.67 5.5 5 y/s Y Flow Direction x r /S 5 ER =.5 5.5 X Z ER =. y/s Flow Direction x r /S Y X Z Streamlines on the planes adjacent to the stepped wall and the sidewall

Z Y Flow Direction Z Y Flow Direction X X y/s y/s Y.5 5 Flow Direction 5 ER =.67.5 5 5 ER =. Z y/s X.5 5 5 ER =.5 Streamlines on the planes adjacent to the flat upper wall and the sidewall

.5 x u /S ER =.67 Nu..7778.5556.3333..8889.6667.4444...5 5 5 ER =..5 x u /S Nu..7778.5556.3333..8889.6667.4444...5 5 5.5 ER =.5 x u /S Nu.3399..7778.5556.3333..8889.6667.4444...5 5 5 Distribution of Nusselt number on the stepped wall

.5 ER =.67 C f..78.56.33.89.44..5 5 x u /S 5.5 ER =. C f..78.56.33..67.44...5 5 x u /S 5.5 ER =.5 C f..56.33.89.44...5 5 x u /S 5 Distribution of friction coefficient on the stepped wall

Effects of aspect ratio Aspect ratio (AR) = WD/(H-S);

Nusselt number contours on the stepped wall (Re = 5)

5 5 6 75 Nusselt number contours on the stepped wall (AR = 6)

Streamwise shear stress component contours on the stepped wall (Re = 5)

Effects of buoyancy force L side wall heated stepped wall Re = g h y x z symmetry center plane Gr = ~ 4 (changing wall heat flux) FLOW H S

Effects of Grashof number on the reattachment region

Boundary of the sidewall recirculating flow region

y/s y/s y/s Flow Direction Flow Direction Step Z Y X 5 x b 5.5 Gr = Step Z Y X 5 x b 5.5 Gr = Step Z Y X 5 x b Flow Direction 5.5 Gr = 4 General flow features at different Grashof numbers

Flow Direction y/s - 4 6 8.5 Flow Direction Re = Gr = Y Z X - y/s 4 6 8.5 Y Re = Gr = Z X Step y/s Step - 4 6 8.5 Flow Direction Y Z X Streamlines on the planes adjacent to the stepped wall, the flat wall and the sidewall Re= Gr = 4 Step

Nu Nu Nusselt number distribution for Re = and Gr =

Conclusions A downwash develops adjacent to the sidewall with a vortex flow that moves in the spanwise direction inside the primary recirculation flow region that is attached to the step; The jet-like developing near the sidewall is responsible for the maximum Nusselt number and the minimum (zero) friction coefficient to develop near the sidewall; The maximum Nusselt number is located near the sidewall, but not at the center width;

Conclusions (Cont.) The two-dimensional flow definition for the reattachment (x u -line), and the limiting streamlines that identify the boundary of the primary recirculation region (x b -line) do not identify the reattachment of this three-dimensional flow; The x u -line increases with Reynolds number within the laminar flow regime, then decreases with increasing Reynolds number. It s change is not obvious for the turbulence regime. The maximum Nusselt number increases with an increase of the aspect ratio and Reynolds number.

Conclusions (Cont.) Increasing the step height increases the reattachment length, the Nusselt number, the size of the sidewall reverse flow region, and the general three-dimensional features of the flow. Increasing the Grashof number results in increasing the Nusselt number and the size of the reverse flow regions adjacent to the sidewall and the flat wall. On the other hand, the size of the primary recirculation flow region adjacent to the stepped wall decreases and detaches partially from the heated stepped wall as Grashof number increases.

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