Similarity solution of axisymmetric non-Newtonian wall jets with swirl

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Abstract

The similarity solution of axisymmetric wall jets with swirl on bodies of revolution for non-Newtonian power-law fluids is presented. The physical and geometrical meaning of all parameters appearing in the course of similarity procedure is treated in detail. Especially, the functional dependence of length, velocity, and pressure similarity scales on the shape and swirl parameters and the power-law flow behaviour index is determined. The already published results related to the similarity solution obtained are discussed.

Introduction

Different types of wall jets are used in mechanical, chemical, and aerospace engineering. For example, they are used for solid surface conditioning associated with heat and/or mass transfer. The knowledge of relevant flow characteristics and governing flow parameters is necessary for studying these transport phenomena. It is worth mentioning that forced or natural convection from a surface of axisymmetric bodies–frequently of arbitrary contour–has been usually investigated through boundary-layer approximation and similarity analysis [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], as well as magnetohydrodynamic flows of electrically conducting fluids [12], [13].

Unlike free (submerged) jets, the flow-structure complexity of wall (submerged) jets arises from the presence of a wall, the inner wall-jet region being significantly affected by the body surface. Glauert [14] was the first to solve the plane and radial wall-jet problem theoretically. He employed boundary-layer equations with adequate wall-jet boundary conditions to obtain a similarity solution for laminar and turbulent flow regimes. The turbulent wall jet (of a Newtonian fluid) in various flow geometries is probably the most investigated wall-jet problem in the past fifty years (e.g. [15] and references therein). A special case of the non-swirling radial wall jet for power-law fluids has been solved by Mitwally [16]. The wall-jet flow of power-law fluids over a curved (both convex and concave) surface has been analysed by Gorla [17]. Wall jets in a stirred tank have been treated by Bittorf and Kresta [18], and Kresta et al. [19]. Adane and Tachie [20] have presented a numerical investigation of three-dimensional wall jet for both Newtonian and non-Newtonian fluids.

This paper presents the similarity solution of swirling wall jets on bodies of revolution for non-Newtonian power-law fluids. The physical and geometrical meaning of all parameters of the similarity solution is discussed in detail. The governing role of parameters characterizing the surface geometry and the rate of rotation is explicitly shown.

Based on the original idea of [14] applied to plane and radial wall jets, the corresponding integral energy equations–dealing with the so-called ‘flux of exterior momentum flux’–have been introduced for the case of wall jets on bodies of revolution in [21] for non-swirling jets, and in [22] for swirling jets. The latter study (dealing with turbulent flow regime) presents a detailed similarity analysis of length, velocity, and pressure scales as functional dependences on the so-called swirl parameter (expressing the rate of rotation) and the shape parameter (characterizing the surface geometry). Filip et al. [23] have shown how to cope with the non-swirling wall-jet flow past axisymmetric bodies for non-Newtonian power-law fluids.

The above-mentioned results [22], [23] can be further extended for the case of swirling wall jets for power-law fluids. It should be noted that in many shear-flow problems numerical solutions and sophisticated flow modelling should be preceded or completed by the similarity analysis revealing analytically the role of relevant flow parameters and their clear physical and geometrical meaning. Moreover, in the present case, the similarity analysis provides a significant simplification of the given problem formulation for further (analytical and/or numerical) calculations.

Section snippets

Problem formulation

Taking into account the axisymmetric shape of the body of revolution, we use the curvilinear coordinate system (x, y, ϕ) with the curvilinear surface coordinate x defined in axial plane according to Fig. 1 where r is a local body radius, rr(x).

The swirling wall jets past axisymmetric bodies for power-law fluids are described by the set of (three) equations of motion (/ϕ0 with respect to axisymmetry) uux+vuyw2r(x)r(x)=1ρτxyy,w2(1r2(x))1/2r(x)=1ρ(Δp)y,uwx+vwy+uwr(x)r(x)=1

Similarity analysis

It is assumed that the flow field is similar, so let us introduce generalized similarity transformations in the form ψ(x,y)=A(x)f(η),η(x,y)=B(x)y(y/δ(x)),w(x,y)=E(x)h(η),τxy(x,y)=ρT1(x)T2(η),τϕy(x,y)=ρT3(x)T4(η),Δp(x,y)p(x,y)p=ρP1(x)P2(η).

Substituting the above similarity transformations into Eqs. (1), (2), (3) we obtain T2+AABr2T1ff+1T1[rA2Br3AABr2A2Br2]f2+rE2rBT1h2=0,P2(1r2)1/2E2rP1Bh2=0,T4+AErT3fh+1T3[AErrAEr2]fh=0 where the primes indicate

Transformation of the original problem formulation

The spatial flow geometry given by (24) is considered as a crucial starting point for introducing the velocity and shear-stress resultants, q and τ respectively, see Fig. 2q=(u2+w2)1/2=ru/ξ=rw/e,τ=(τxy2+τϕy2)1/2=rτxy/ξ=rτϕy/e, where ξξ(x)=(r2(x)e2)1/2, and for introducing a differential element dζ in the resulting flow direction past the axisymmetric body surface (ζ may be considered as the curvilinear surface coordinate following the resulting ‘helical’ fluid motion past the body surface,

Similarity solution

The solution is sought in the similarity form ψ̄(ζ,y)=Ā(ζ)f̄(η̄),η̄(ζ,y)=B̄(ζ)y(y/δ̄(ζ)) where the stream function ψ̄ fulfils q=ξ1ψ̄/y,v=ξ1ψ̄/ζ.

The transformations (25), (26), (27), (28), (29) are exclusively x-dependent, consequently the universal transverse similarity structure of the velocity field remains unchanged, so f̄f. Numerical results for the similarity function f are obtained for the corresponding similarity equation and boundary conditions, namely n|f|n1f+ff+Cf2=

Discussion

The integral quantity W represents a specific wall-jet flow invariant which can be obtained within the frame of similarity analysis only (cf. [23]). This quantity which may be understood as a ‘generalized flux of momentum flux’, serves as a certain substitution for the Glauert-type integral invariant dealing with the ‘flux of exterior momentum flux’ 0ρξ(x)q(yρξ(x)q2dy)dy=const which is valid exclusively for the wall-jet flow of a Newtonian fluid, n=1.

The obtained solution (43), (44), (45),

Conclusions

The similarity solution of axisymmetric wall jets with swirl on bodies of revolution for non-Newtonian power-law fluids has been presented. The analytical expression for the length, velocity, and pressure similarity scales has been explicitly determined. All the relevant parameters–shape and swirl parameters, power-law model parameters, integration constants and integral conditions–appearing in the similarity solution obtained possess a specific physical and geometrical meaning which should be

Acknowledgements

This work was supported by the Grant Agency of the Acad. of Sci. of the Czech Rep. through grant IAA200600801, and by the Acad. of Sci. of the Czech Rep. through AV0Z20600510.

References (29)

  • S. Roy et al.

    Unsteady laminar compressible swirling flow with massive blowing

    AIAA J.

    (1992)
  • A. Nakayama et al.

    Buoyancy-induced flow of non-Newtonian fluids over a non-isothermal body of arbitrary shape in a fluid-saturated porous medium

    Appl. Sci. Res.

    (1991)
  • A. Nakayama et al.

    General similarity transformation for combined free and forced-convection flows within a fluid-saturated porous medium

    J. Heat Transfer, Trans. ASME

    (1987)
  • J.L.S. Chen

    Mixed convective flow about slender bodies of revolution

    J. Heat Transfer, Trans. ASME

    (1987)
  • Cited by (4)

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