Wuhan, , P.R. China Published online: 25 Jan 2007.

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1 This article was downloaded by: [Huazhong University of Science & Technology ] On: 26 December 2014, At: 20:25 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK International Journal of Comutational Fluid Dynamics Publication details, including instructions for authors and subscrition information: htt:// An integrated 2-D Navier Stokes equation and its alication to 3-D internal flows A. Nakayama a, F. Kuwahara a & W. Liu b a Shizuoka University, Deartment of Mechanical Engineering, Johoku, Hamamatsu, , Jaan b School of Energy and Power Engineering, Huazhong University of Science and Technology, Wuhan, , P.R. China Published online: 25 Jan To cite this article: A. Nakayama, F. Kuwahara & W. Liu (2006) An integrated 2-D Navier Stokes equation and its alication to 3-D internal flows, International Journal of Comutational Fluid Dynamics, 20:2, , DOI: / To link to this article: htt://dx.doi.org/ / PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the Content ) contained in the ublications on our latform. However, Taylor & Francis, our agents, and our licensors make no reresentations or warranties whatsoever as to the accuracy, comleteness, or suitability for any urose of the Content. Any oinions and views exressed in this ublication are the oinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied uon and should be indeendently verified with rimary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, roceedings, demands, costs, exenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and rivate study uroses. Any substantial or systematic reroduction, redistribution, reselling, loan, sub-licensing, systematic suly, or distribution in any form to anyone is exressly forbidden. Terms & Conditions of access and use can be found at htt://

2 International Journal of Comutational Fluid Dynamics, Vol. 20, No. 2, February 2006, An integrated 2-D Navier Stokes equation and its alication to 3-D internal flows A. NAKAYAMA *, F. KUWAHARA and W. LIU Deartment of Mechanical Engineering, Shizuoka University, Johoku, Hamamatsu , Jaan School of Energy and Power Engineering, Huazhong University of Science and Technology, Wuhan , P.R. China Nomenclature An efficient two-dimensional (2-D) analytical and numerical rocedure has been roosed to investigate three-dimensional (3-D) internal flows through a assage with a satially variable deth, in which the viscous forces act significantly on both uer and lower walls. The integral 2-D version of the Navier Stokes equation was obtained by integrating the full Navier Stokes equation in a 3-D form over the deth of the assage. In order to examine the validity of the integrated momentum equations, fully-develoed flows in straight noncircular ducts were investigated analytically rior to numerical investigations. It has been shown that the exact solutions for circular, ellitical and equilateral triangular ducts are obtainable from the integrated Navier Stokes equation. Having confirmed its wide alicability to internal flows, numerical comutations were conducted to investigate the oscillation mechanism of a fluidic oscillator. Comarison of the resent rediction and exeriment reveals the validity of the resent treatment. Keywords: Internal flow; Numerical method; Fluidics; Oscillation; Non-circular ducts f velocity rofile function h(x, y) half deth of the assage, function for the wall geometry, u, v, w velocity comonents in the x, y and z directions u; v average velocity comonents u in bulk mean velocity at the nozzle inlet ressure Re Lin ¼ u inl in n Reynolds number based on L in and u in St ¼ fl in u in Strouhal number t time x, y, z Cartesian coordinates n kinematic viscosity r density z ¼ dimensionless coordinate z hðx;yþ Subscrit in 1. Introduction nozzle inlet Internal flows encountered in engineering alications are three-dimensional (3-D) in nature. Hence, we often have to aeal to 3-D comutations in order to cature the details of velocity and ressure fields for designing heat and fluid flow equiment. Such 3-D comutations, however, are extremely exensive and time consuming, even when only steady sate solutions are required. Ironically, we may find it difficult to extract meaningful flow characteristics directly from the results obtained in such 3-D comutations, because of excess satial information. Thus, having comleted full 3-D comutations, the 3-D results are often integrated over a certain coordinate to reduce to the averaged values. Such averaged quantities are much easier to areciate, making it ossible for us to draw the overall asects of 3-D flow characteristics much better. Recent advances in microfabrication technologies have been so romising that some micro-fluidics may comete with conventional mechanical and electrical systems. Fluidic devices such as feedback fluidic oscillators (e.g. Trietts et al. 1973, Parry et al. 1991, Zemel and Furlan 1996) and fluidic flowmeters (e.g. Boucher and Mazharoglu 1988, Mansy and Williams 1989, Lua and Zheng 2003) consist of comlex 3-D assages with a small deth. The foregoing consideration romts us to introduce an efficient two-dimensional (2-D) numerical calculation rocedure for a 3-D internal flow through a assage with a satially variable deth, in which the viscous forces act *Corresonding author. tmanaka@ic.shizuoka.ac.j International Journal of Comutational Fluid Dynamics ISSN rint/issn online q 2006 Taylor & Francis htt:// DOI: /

3 100 A. Nakayama et al. significantly on both uer and lower walls. We shall derive such a set of 2-D governing equations by integrating the full 3-D Navier Stokes equation over the deth. The resulting integrated momentum equations will be solved for the deendent variables, namely, the mean velocity comonents and ressure averaged locally over the deth, which varies satially. An analytically reduced version of the integrated momentum equation is alied for the roblems of fullydeveloed flows in noncircular ducts, so as to restore a class of exact solutions, which are known for noncircular shaes. In this way, the soundness of the integrated momentum equation is elucidated. Furthermore, this set of the integrated momentum equations is discretized using a finite volume method, so as to conform with SIMPLE algorithm. A numerical exeriment is conducted to investigate the oscillation mechanism of a feedback fluidic oscillator, which is designed to sray water to an automobile windshield. The frequency redicted by the resent numerical rocedure is in good accord with that of the exeriment, which roves the validity of the resent economical numerical rocedure. 2. Two-dimensional Navier Stokes equation integrated over the deth We shall consider comlex assages such as used in fluidic oscillators for sraying water to the automobile windshield, as shown in figure 1. We assume that the assage is shallow enough for the flow within the assage to stay laminar. The governing equations in consideration, namely, the continuity, Navier Stokes and energy equations for incomressible flows are given by u v w z ¼ 0 ð1þ u u 2 2 n u vu 2 n u wu 2 n u z z ¼ 2 1 r v uv 2 n v v 2 2 n v wv 2 n v z z ¼ 2 1 r w uw 2 n w vw 2 n w w 2 2 n w z z ð2þ ð3þ ¼ 2 1 r z : ð4þ The assage in consideration is symmetric with resect to the x y lane such that the uer and lower wall geometries are given by z ¼ ^h(x, y), resectively. The following rocedure aears to be similar to that of Hele Shaw flow. However, it is noted that h(x, y) can vary satially, and that both inertial and viscous terms are retained. We integrate the continuity equation from 0 to h(x, y) with resect to z as: ð h dz þ 0u ð h v dz ¼ 0: 0 We assume the velocity rofiles are: uðt; x; y; zþ ¼uðt; x; yþf ðzþ and ð5þ where vðt; x; y; zþ ¼vðt; x; yþf ðzþ z ¼ z hðx; yþ ð6þ ð7þ and uðx; yþ and vðx; yþ are the velocity comonents averaged over the deth, such that the symmetric function f(z) should satisfy Figure 1. Fluidic oscillator used for sraying water. f ð^1þ ¼0 and f 0 ð0þ ¼0: ð8þ

4 The foregoing integral form of the continuity equation (5) may be rewritten as Navier Stokes equation and its alication 101 uh vh ¼ 0: ð9þ Since w(x, y, z). 0, the z-momentum equation (4) immediately gives ¼ (t, x, y). We substitute the velocity rofiles given by equation (6) and ¼ (t, x, y) into the x- and y-momentum equations (2) and (3), and then integrate them over the deth to find h u hu 2 2 n hu hvu 2 n hu ¼ 2 h r þ n f 0 ð1þ u h 1 h 2 h! 2 h v huv 2 n hv ¼ 2 h r þ nf 0 ð1þ v h 1 h hv 2 2 n hv 2 h! 2 ð10þ : ð11þ One of the simlest candidates for the symmetric function f(z) is assumed as: f ðzþ ¼ 3 2 ð1 2 z 2 Þ ð12þ such that f 0 (1) ¼ 23. The rofile given by the function conforms to the no-sli and symmetry conditions given by equation (8). The integrated momentum equations (10) and (11) along with the integrated continuity equation (9) form a comlete set of the governing equations for a 3-D internal flow through a comlex assage with a small deth. These governing equations subject to no-sli conditions are believed to be valid for all assages described by a moderately varying arbitrary function h(x, y). 3. Fully develoed flows in noncircular ducts We shall examine the validity of the integrated momentum equations, considering fully develoed flows in straight noncircular ducts, as shown in figure 2. For the case of fully develoed flows, the integrated set of the equations reduce to the following ordinary differential equation in a dimensionless form as: dy where d 2 *2 h* u* 2 3 u* h* 1 þ dh*! 2 þ h* ¼ 0 dy* ð13þ Figure 2. The duct width may be chosen for the reference length, as shown in figure 3. As the duct shae h*( y*; 0 # y* # 1) is given, the foregoing second order O.D.E. may be solved for u* ð y* Þ using the obvious boundary conditions, namely, u* ð0þ ¼u* ð1þ ¼0: ð15þ Instead of secifying a articular duct shae, h*( y*), to find its vertically averaged velocity rofile u* ð y* Þ, we shall take an inverse solution rocedure. That is to find a family of ossible duct shae functions h*( y*), uon assuming the velocity rofile function in the form of u*ð y*þ ¼cy*að1 2 y*þ b ð16þ such that it automatically satisfies the boundary conditions given by equation (15). Substitution of the forgoing velocity function (16) into equation (13) and some maniulation lead to u*ð y*þ ¼cy*að1 2 y*þ b ¼ 2 1 h* Coordinate system for a straight noncircular duct. d 2 h* dy *2 þ 2 a y* 2 b y* h* aða 2 1Þ 2ab þ 2 y *2 y*ð1 2 y*þ 2 3 h*2 1 þ dh*!! 2 21 : dy* dh* dy* þ bðb 2 1Þ ð1 2 y*þ 2 ð17þ We note the constancy of the exonents a and b in the foregoing exression, and find the family of ossible duct shae functions, namely, h* ¼ð1= ffiffiffi 3 Þy* and h* ¼ g ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y*ð1 2 y*þ, which corresond to the cases of u u* ; ; h* ; h and y* ; y : 2 L2 ref m d dx ð14þ Figure 3. Cross-sectional view of noncircular duct.

5 102 A. Nakayama et al. rofile of Hagen Poiseuille flow, namely, uðrþ ¼ d ðr 2 2 r 2 Þ m dx ð22þ where R ¼ =2 and r 2 ¼ðy 2 RÞ 2 þ z 2 : ð23þ Figure 4. Duct cross-sections: (a) Equilateral triangular cross-section; (b) Ellitical cross-section. The foregoing reliminary consideration suggests a wide range of alicability of the resent integral momentum equations. equilateral triangle and ellitical sections, resectively, as shown in figure 4(a) and (b). Equilateral triangle of side 2 = ffiffi 3 : u* ¼ 1 6 y*2 ð1 2 y*þ for h* ¼ 1 ffiffi y* 3 and ellitical section with axes and g : u* ¼ g 2 3ð1 þ g 2 y*ð1 2 y*þ Þ for h* ¼ g ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y*ð1 2 y*þ: ð18þ ð19þ These dimensionless functions for the average velocity can be translated to find the local velocity field over the duct cross-section uð y; zþ ¼ 2 L2 ref m y z u* f h d dx as equilateral triangle of side 2 = ffiffi 3 : uð y; zþ ¼ L2 ref d m dx y y! z 2 and ellitical section with axes and g : uð y; zþ ¼ g 2 2ð1 þ g 2 Þ y 2 L ref d m dx 1 2 y 2 z! 2 ð20þ ð21þ which turn out to be the exact solutions obtainable directly from Navier Stokes equation (e.g. White 1974). Equation (21) for the case of g ¼ 1 reduces to the well-known 4. Numerical simulation of jets from a fluidic oscillator In what follows, the fluidic oscillator with constant h, as already shown in figure 1, will be examined numerically, using the integrated momentum equations (10) and (11) along with the integrated continuity equation (9). This oscillator is one of tyical fluidic devices installed in some Jaanese automobiles for sraying water to the automobile windshield. The numerical results obtained here are comared with the exerimental data recently conducted by Kuwahara et al. (2005) so as to investigate the validity of this economical numerical rocedure and its acquired accuracy. All governing equations conform to the following general transort equation: h f hf huf 2 G f hf hvf 2 G f ¼ S f ð24þ which was then integrated over a small element and time interval to form a general discretized equation for the oint P and its neighboring oints E, W, N and S, as follows: a P f P ¼ a E f E þ a W f W þ a N f N þ a S f S þ b: ð25þ The well-known QUICK scheme has been adoted for differencing the advection terms. Then, they were numerically solved using SIMPLE algorithm roosed by Patankar and Salding (1972). Convergence was measured in terms of the maximum change in each variable during an iteration. The maximum change allowed for the convergence check was set to 10 25,as the variables are normalized by aroriate references. Further details on this numerical rocedure can be found in Patankar (1980) and Nakayama (1995). All comutations were erformed using the comuter system at Shizuoka University Comuter Center. In this study, the Cartesian grid system is used instead of the body fitted system. The latter is recommended if the detailed flow field within the assage needs to be exlored. However, when only macroscoic characteristics such as

6 Navier Stokes equation and its alication 103 Figure 5. Grid nodes within the fluidic oscillator. oscillation frequency are needed, the former would suffice for the urose. Geometrical data from a CAD are first translated in a set of bm data, which is then fed into the re-rocessor develoed by our grou to generate a grid system automatically. A tyical grid system consists of nodes with dense and coarse meshes for the regions inside and outside of the assages, resectively, to cover a large domain of integration, including both the fluidic devise and its surroundings, namely, mm 2. Figure 5 shows a art of dense mesh distributed around the fluidic oscillator for the resent non-uniform grid system. Preliminary calculations were made to comare the results against those obtained with nodes for some selected cases. In this way, the originally used grid resolution was found sufficient. Moreover, the time ste was set small enough to satisfy Courant condition, after confirming that any further decrease in the time ste does not alter the results significantly. Numerical comutations were initiated setting a uniform velocity u in at the nozzle inlet, which was varied from 2.2 to 6.0 m/s, at the inlet. For this oeration range, the Reynolds number Re Lin based on the inlet width L in and velocity u in varies from 4400 to 12,000. This is the range in which Kuwahara et al. (2005) reviously conducted an exeriment to measure the frequency of oscillating flow by a stroboscoe. It takes about 10 cycles of oscillations for the velocity to attain its eriodically fully-develoed stage. Figure 6 (a) (f) deict a comlete cycle of the eriodically fully-develoed velocity field for the case of Re Lin ¼ These figures clearly show that the oscillation frequency for this case is about f ¼ 1/2.22 ms ¼ 450 Hz. The results of comutations carried out for the fluidic oscillator were assembled in terms of Strouhal number St ; fl in /u in. The redicted Strouhal number St stays almost constant, namely, St ; fl in /u in. 0.4, over the range of Reynolds number studied, which turns out to be quite close to the exerimentally observed value. The mechanism of the fluidic oscillation is quite comlex, as can be seen from these figures. Another set Figure 6. Oscillating velocity field.

7 104 A. Nakayama et al. of comutations were carried out deleting five circular cylinder obstacles located ustream of the round equilateral triangular cavity. Oscillations were observed just as in the case of the assage with the obstacles. Thus, it is not the ustream flow obstruction but the imbalance in two jets meeting each other within the round equilateral triangular cavity that causes such fluidic oscillations. A arametric study can readily be made to find ossible geometrical arameters to control its oscillation attern and frequency, since the interface between the resent CFD code and CAD system has been already established. However, such an attemt is out of the scoe of the resent study. 5. Concluding remarks Full Navier Stokes equation in the 3-D form was integrated over a local channel deth to obtain the integrated 2-D Navier Stokes equation for analyzing 3-D internal flows through a assage with a satially variable deth. Firstly, the integrated momentum equation was alied for the roblems of laminar fully-develoed flows in noncircular ducts, so as to restore a class of exact solutions, namely those for circular, ellitical and equilateral ducts. Secondly, a numerical comutation code was develoed and couled with a CAD system, so as to conduct a numerical exeriment, demonstrating its effectiveness for designing fluid flow devices. The numerical rocedure develoed in this study can easily be extended to the case of turbulent flows. Some friction law should be introduced to evaluate the turbulent frictional forces on the uer and lower walls. Such investigations are underway. Acknowledgements The authors would like to exress their sincere thanks to President Morikawa, Technical Division Head Kageyama and Senior Researcher Tsukamoto of Nihon Biniron Co., Hamamatsu, Jaan, for their suort. References Boucher, R.F. and Mazharoglu, C., Low Reynolds number fluidic flowmetering. J. Phys. E: Sci. Instrum., 1988, 21, Kuwahara, F., Okazaki, S and Nakayama, A, A three-dimensional internal flow through a comlex assage with a small deth. 42nd National Heat Transfer Symosium of Jaan, 2005, III, Lua, A.C. and Zheng, Z., Numerical simulations and exerimental studies on a target fluidic flow meter. Flow Meas. Instrum., 2003, 14, Mansy, H. and Williams, D.R., An exerimental and numerical study of traed vortex air fluidic flowmeter. ASME FED Forum on Turbulent Flows, 1989, 76, Nakayama, A., PC-Aided Numerical Heat Transfer and Convective Flow, 1995 (CRC Press: Boca Raton). Parry, A.J., Chiwanga, S.G., Kalsi, H.S. and Jeson, P., Numerical and exerimental visualization of flow through a target fluidic oscillator. ASME FED Exerimental and Numerical Flow Visualization, 1991, 128, Patankar, S.V. and Salding, D.B., A calculation rocedure for heat, mass and momentum transfer in three-dimensional arabolic flows. Int. J. Heat Mass Transfer, 1972, 15, Patankar, S.V., Numerical Heat Transfer and Fluid Flow, 1980 (Hemishere: Washington, DC). Trietts, J.R., Ng, H.K. and Royle, J.K., An oscillating bistable fluid amlifier for use as a flowmeter. Fluidics Q., 1973, 5(1). White, F.M., Viscous Fluid Flow, , 1974 (McGraw-Hill Book Comany: New York). Zemel, J.N. and Furlan, R., Microfluidics, Handbook of Chemical and Biological Sensors, edited by R.F. Taylor and J.S. Schultz, , 1996 (Institute of Physics Publishing: Bristol and Philadelhia).