- Email: [email protected]

Interaction of a surface mounted hot body with a turbulent boundary-layer J.L. Carvalho, A.R.J. Borges* Mechanical Engineering Department, Faculdade de CieL ncias e Tecnologia, Universidade Nova de Lisboa, 2825 Monte de Caparica, Portugal

Abstract The main objective of this study is to understand the flow structure close to the roughness elements over which a thick turbulent boundary-layer is developed and the way it interacts with a surface mounted hot cube having in mind possible applications to the thermal interaction of a building located in a urban area under light wind or calm atmospheric conditions. The whole structure of the turbulent boundary-layer is studied in what concerns mean flow field, turbulent intensity and shear stress profiles. Boundary-layer integral parameters are calculated and their along wind development is studied, in a relative wide range of different roughnesses. Different flow conditions are used, ranging from natural to forced convection. Particular attention is given to mixed convection situations typical of a wide variety of practical problems such as building environment and industrial equipment. The influence of wind incidence is also studied. The study comprises also a numerical simulation, using a three-dimensional computer code based on the k—e two equation turbulence model. Numerical results are compared with experimental ones in what concerns the development of the turbulent boundary layer itself and the way it interacts with the heated bluff body. ( 1998 Elsevier Science Ltd. All rights reserved. Keywords: Turbulence; Boundary-layer; Heat-transfer; Thermal plumes

1. Introduction Heavy built up urban areas create local conditions that, from aerodynamic point of view, can be modelled as the development of a turbulent boundary-layer over a very rough surface. Such a flow influences the pressure distribution on building’s surfaces and convection heat transfer from them in a rather complex way [1]. As compared

* Corresponding author. E-mail: [email protected] 0167-6105/98/$19.00 ( 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 6 1 0 5 ( 9 8 ) 0 0 0 4 3 - 9

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with the classical Nikuradse problem [2] (the flow over a compact layer of sand grains with diameter d glued to a solid surface, over which a turbulent boundary-layer 4 develops), the flow over a regular distribution of discrete elements, compact enough to assure maximum wall shear stress [3], leaves a distance between them, ¸ of the order d , that somehow preserves the aerodynamics of the individual elements. 4 The flow between the elements is highly turbulent and its structure assures the shear stress at the surface; it is obviously dependent on the parameters b/¸ and h/¸, b and h being the element width and height respectively. In the cases studied the above mentioned geometrical ratios are of the order of one and such a flow can roughly be described as an horse shoe vortice structure wrapping around each element [4]. This mechanism derives from the very large mean shear present, associated with pressure suction on the top and the rear faces of the roughness elements. In this paper a study of the flow over staggered distribution of cubes, having a zero pressure gradient is made. As shown measured values of drag and heat convection coefficients can be explained considering local mean speeds slightly below the top of the elements. Although present results have been obtained with a small scale model (the scale being of the order of 1 : 1000) it is thought that the nondimensional correlations established set up rationale that can be tentatively used at full scale.

2. Approaching flow 2.1. Experimental set-up All experimental studies were carried out using a boundary-layer wind tunnel of the close circuit type with a test section 1]1.5 m2 and 9 m long. Air speed can be continuously varied from 0 to above 30 m/s. The longitudinal pressure gradient can be adjusted by vertically displacing the test chamber ceiling. The roughness was created by a staggered arrangement of cubes mounted in the test chamber floor with ¸"2b. Boundary layer flow characteristics were varied by simulating different heights (0(h(30 mm) of the roughness elements. This was done by adjusting the vertical position of the slabs existing between each pair of transversal rows of cubes. 2.2. Along wind development In order to know the characteristics of the approaching flow, mean flow velocity, turbulence intensity and shear stress profiles were measured by traversing a Pitot tube and hot-wire probes. The integral parameters of the boundary-layer: thickness (d), displacement thickness (d ), momentum thickness (d ) and the shape factor $ . (H"d /d ) were then calculated. $ . Assuming that the mean velocity distribution can be represented by the simple power law

AB

º y [email protected] , " º d =

(1)

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477

where º is the free stream velocity and y is the vertical distance to the surface, and = using the relation º ="C [email protected](n`1), 1 d u*

(2)

where u*"Jq /o is the wall-friction velocity (q being the wall shear stress) and 8 8 Re "º d/l, we can obtain simple relations for d, d and d as a function of x, the d = $ . longitudinal cartesian coordinate with its origin at the test chamber entrance [5]. In fact, from Eq. (1) and the definitions of d, d , d and H we arrive at $ . d 1 d n n#2 $" ." , and H" . (3) d d n#1 (n#1)(n#2) n Defining now the friction coefficient in the usual way as c "q /1oº2 we obtain & 82 = c u* 2 &" . (4) 2 º = From von Ka´rma´n momentum integral equation for flows with zero longitudinal pressure gradient, and according to Eqs. (2) and (4)

A B

A B

dd º ~2 q ." 8 " = "[email protected](n`1). 1 d dx u* oº2 = Relating Eq. (5) with Eq. (3) and integrating we finally obtain

C

(5)

D

(n`1)@(n`3) º d º = ." C = (x!x ) 2 l 0 l where

C

D

[email protected](n`1) n#3 n C "C~2 . 1 n#1 (n#1)(n#3) 2

(6)

This is the theoretical relation between d and x, x being the x value at the . 0 boundary-layer virtual origin. Similar expressions were obtained for d and d . Cor$ relating experimental data, we can additionally estimate the turbulent boundary-layer virtual origin, x , for any case studied. 0 Shape factor H along wind evolution is represented in Fig. 1 indicating a quasi-fully developed flow at the end of the test chamber. Lines represent the asymptotic values of H calculated using the empirical function H"(1!6.3(u*/º ))~1 [5], based on = experimental data for smooth and rough plates [6]. Also represented are the integral parameters d, d , and d based on the above mentioned theoretical functions. Having $ . used different surface roughnesses, we obtained different roughness heights, y , vary0 ing from 2 to 8 mm. Shadowed regions drawn in Fig. 1 indicate correspondent ranges for integral parameters d, d and d . The figure additionally shows that x is always $ . 0 negative.

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Fig. 1. Along wind development of boundary-layer integral parameters.

Experiments were then conducted at a fixed test section (x"7.27 m) where quasifully developed flow was obtained. The body used to generate the dynamic perturbation of the flow is a cube equal in size to the roughness elements but mounted on a small horizontal turning plate slightly above the tunnel floor at a level d, the virtual origin of the boundary-layer, according to the logarithmic law

A B

º 1 y!d " ln , (7) u* s y 0 where y is the roughness height and s is the von Ka´rma´n constant. This cube was 0 made out of aluminium plate 1.5 mm thick, provided with small pressure taps conveniently located in order to measure pressure distribution with sufficient accuracy. 2.3. Mean velocity and turbulence intensity profiles Fig. 2 shows typical mean flow and turbulence intensity profiles for both minimum and maximum roughnesses used. Particular attention must be given to the extremely high level of turbulence intensity in the inner region of the boundary-layer, specially noticeable when we base the turbulence intensity on local velocity, º, instead of free stream velocity, º . = Wall shear stress measurements were made using a slant wire constant temperature probe, the results being correlated with those derived from von Ka´rma´n momentum integral equation. Additionally wall shear stress was determined after computing roughness drag calculated by integrating pressure coefficients over a test cube with a surface equipped with pressure taps connected to a micromanometer. Having covered a range of roughnesses and Reynolds numbers all results show good agreement between them for the mentioned independent measuring techniques. This was reported elsewhere [7].

J.L. Carvalho, A.R.J. Borges/J. Wind Eng. Ind. Aerodyn. 74—76 (1998) 475–483

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Fig. 2. Typical mean velocity and turbulence intensity profiles for (a) y "2 mm; (b) y "8 mm. 0 0

3. Thermal interaction with the flow 3.1. Experimental technique In heat transfer experiments the above mentioned aluminium cube was substituted by another machined out of solid copper and surface mounted as already described. Its temperature could be varied by changing electric current supplied to an internally mounted electrical cartridge. Temperature of the cube surfaces ranging from 140°C to 160°C were thereby obtained for different wind speeds. In this way isothermal conditions at the cube faces were obtained. Careful insulation was provided under the bottom surface of the cube in order to make those heat losses negligible. Cube faces interacting with the flow were kept polished and temperature limited (160°C) in order to minimise radiation losses. 3.2. Nusselt numbers from natural to forced convection Fig. 3 shows average Nusselt number dependence with Reynolds number for wind incidence at 0° and at 45°. The figure associates present results with earlier measurements by Quintela [8], using a different set-up and other thermal conditions. Also included are the asymptotic values of Nu when Re equals zero and we reach natural convection regimen. When in forced convection, both experimental results compare well for wind incidence at 0° with those deducted from a mass transfer study by Natarajan and Chyu [9].

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Fig. 3. Nusselt number dependence with Reynolds number for wind incidence at 0° and 45°.

Nusselt number, Nu"hD/k, was defined in the usual way, taking the side of the cube as typical length and calculating the values of air heat conduction coefficient k at film temperature. Reynolds number, Re"ºD/l, usually takes the free stream velocity, º , as an = adequate reference speed. This is the case in the above mentioned studies [8,9] where

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approaching flow was a thin turbulent boundary-layer (d(D). The present work deals with a thick boundary layer (d+8D) and a reference speed taken at a level y"0.75D was used. This reference speed proved to be of general validity when we study the interaction with a surface mounted body. Interesting was to verify that such level approximates the level where stagnation point occurs. Since Re is defined in such a way, present results exhibit reasonable good agreement with other results in all regimens studied. Furthermore, results indicate a week dependence of Nusselt number on angular position of the test cube. The same was previously verified with the drag coefficient [7]. This was to be expected having in mind the very high intensity of turbulence prevailing near the cube at low levels in the boundary-layer. A lack of experimental results in lower range of Re is due to the difficult in simulating mixed convection. We must have low velocities, frequently below the limits of anemometer equipment, and high temperatures creating additional difficulties related with quantification of radiation and conduction losses through the bottom face of the cube.

4. Numerical results 4.1. The turbulence model A computer code developed by Delgado [10] concerning the numerical simulation of three-dimensional time averaged Navier—Stokes equations by means of a k—e two equation turbulence model was used applying boundary conditions suitable to the flow around bluff bodies. In this code an hybrid interpolation scheme, combining central and upwind finite differences, is applied to a staggered grid [11]. To enforce continuity the finitedifference equations were solved iteratively using the SIMPLE algorithm of Patankar and Spalding [12]. Standard values for the empirical constants include in the k—e turbulence model were used except for p , the turbulent Prandtl number, and C , the 5 3 empirical constant that affects the buoyant term in the e-equation. The value p "0.7 5 was adopted after a parametric study in order to obtain the best correlation between numerical and experimental results. C was made equal 0.8 according to Hossain 3 [13]. Near solid surfaces velocity and temperature were specified through the use of wall-function method. 4.2. Numerical and experimental results Numerical results agree well with experimental ones when we simulate the development of the turbulent boundary layer over the wind tunnel rough wall with different roughness and flow conditions. Also reasonable good agreement was obtained when simulating natural convection around the heated cube, Fig. 4. When in mixed or forced convection, Fig. 5, considerable differences were obtained when we tried to simulate the flow around the cube with or without thermal effects.

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Fig. 4. Thermal plume on natural convection (Re/JGr"0). ¹ !¹ "120°C and y "8 mm. #6"% = 0

Fig. 5. Thermal plume on mixed convection (Re/JGr"3). Central plane isothermal lines of ¹!¹ for = ¹ !¹ "135°C and º "1 m/s; y "8 mm. #6"% = = 0

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A possible explanation for the significant differences might be related to the first order upwind finite difference scheme included in the algorithm, which promotes somehow a numerical and then false diffusion. Delgado [14] suggests a higher order method to improve the accuracy of the results without excessive grid density. In the near future the actual computer program will be updated along these lines. 5. Conclusions A study of the interaction of a surface mounted hot cube with a turbulent boundary-layer developing on a very rough surface was made. Both experimental and numerical results were presented. Experimental results were presented as Nusselt number dependence on Reynolds and Grashof numbers. Present results clearly correlate with previous results obtained with thin boundary layers and much less turbulence levels, provided we establish as reference flow speed the value existing at 3/4 of the cube height. Not surprisingly, present results show much less dependence of Nusselt number on wind incidence angle on the cube than those reported by previously mentioned authors. Numerical results reveal their usefulness as a rational description of the flow but inevitably show their serious limitations that can possibly be due to the structure of turbulence model adopted and the false diffusion generated by the scheme of interpolation used. References [1] D.A. Quintela, D.X. Viegas, Convective heat losses from buildings, in Wind Climate in Cities, NATO ASI Series. E 277 (1995) 503—522. [2] H. Schlichting, Boundary—layer theory, McGraw—Hill Book Co, USA, 1968. [3] J. Counihan, Wind tunnel determination of the roughness length as a function of the fetch and the roughness density of three-dimensional roughness elements, Atmos. Environ. 5 (1971) 637—642. [4] P.R. Owen, W.R. Thomson, Heat transfer across rough surface, J. Fluid Mech. 15 (1963) 321—334. [5] J.O. Hinze, Turbulence, 2nd ed., McGraw-Hill Book Co, USA, 1975. [6] F.R. Hama, Soc. Naval Architects Marine Engrs. Trans. 62 (1954) 333. [7] A.R.J. Borges, J.L. Carvalho, Aerodynamic and thermal interaction of an element of a very rough surface with a turbulent boundary—layer flow, EECWE’94 Preprints, Warsaw, Poland, 1 (1994) pp. 71—77. [8] D.A. Quintela, Convecc7 a8 o te´rmica a partir de um corpo cu´bico imerso numa camada limite turbulenta, Ph.D. Thesis, University of Coimbra, Portugal, 1989. [9] V. Natarajan, M.K. Chyu, Effect of flow angle-of-attack on the local heat/mass transfer from a wall—mounted cube, Trans. ASME 116 (1994) 552—560. [10] J.D. Delgado, Contribuic7 o8 es para o estudo da ventilac7 a8 o natural de edifı´ cios. Ph.D. Thesis, Universidade Nova de Lisboa, Portugal, 1989. [11] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Corporation, Washington, 1980. [12] S.V. Patankar, D.B. Spalding, A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows, Int. J. Heat Mass Transfer 15 (1972) 1787—1806. [13] M.S. Hossain, Mathematische modellierung von turbulenten auftriebsstromungen, Ph.D. Thesis, University of Karlsruhe, 1979. [14] J.D. Delgado, A numerical simulation of the flow patterns inside a building due to wind action and heat release, EECWE’94 Preprints Warsaw, Poland, 1 (1994) pp. 159—168.