FEDSM PRESSURE DISTRIBUTION ON THE GROUND BY IMPINGING TWO-DIMENSIONAL JET DUE TO A VORTEX METHOD

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1 Proceedings of the 3rd SME/JSME Joint Fluids Engineering Conferences July 8-23, 999, San Francisco, California, US FEDSM PRESSURE DISTRIBUTION ON THE GROUND BY IMPINGING TWO-DIMENSIONL JET DUE TO VORTEX METHOD Teruhiko Kida Fluid Dynamic Engineering Laboratory Department of Energy Systems Engineering Osaka Prefecture University Sakai, Osaka Takanori Take Department of Mechanical Systems Engineering Shiga Prefecture University Hikone, Siga Masatake Toshima 2 Department of Energy Systems Engineering Osaka Prefecture University Sakai, Osaka Mituo Kurata Department of Mechanical Engineering Setunan University Neyagawa, Osaka BSTRCT The main purpose of the present paper is to show the method how to obtain the pressure distribution for impinging two-dimensional ets from the flow behavior obtained by a vorte method. The typical application of the system of two impinging ets is the floater: It is formed an air cushion chamber by the impinging ets and supports some heavy flat plates made from various materials. To increase the efficiency of the floater, it is necessary to know the detailed flow behavior in the cushion chamber and the ets. In the present paper, a vorte method is used to simulate two-dimensional impinging ets to know the mechanism of the formulation of the cushion chamber, however, it is one of difficult problem to obtain the pressure distribution from numerical results of the vorte method. The general approach is formulated to obtain the pressure distribution for the vorte blob method with Gaussian cut-off function and its approach is applied to the problem of the impinging et. Further, it is shown that the unsteady Bernoulli equation is applicable to high Reynolds number flows to obtain the pressure distribution. Gakuen-cho -, Sakai, Osaka , Japan. 2 Present address: Torishima Pump, Takatuki, Osaka , Japan. INTRODUCTION Floating systems are widely used in the field of the transfer of untouchable materials: In the heat treatment for aluminum or organic film, they are used to increase the quality assurance to prevent scratches on the surface of strips. Recently, there are many cases operating under high temperature for annealing of silicon steel or stainless steel. In these cases, the temperature of air becomes up to Cor more over and it is necessary to be cooling down the temperature: The floater uses the heat resistant casted alloy or has a cooling water duct. In particular, the temperature of air must be cooled when it passes through air supply such as a fan. Thus, to support strips efficiency is to save the energy of air supply, that is, the save of mass flow flu supporting the strips is important. Therefore, it is necessary to know the detail of flow behavior of et impinging to the ground and also global characteristics of the impinging et. The purpose of the present paper is to know the mechanism of the cushion chamber between two impinging ets and to know the effect of shape of the nozzle. In particular, we have to know whether or not the pressure drop in Copyright c 999 by SME

2 the cushion chamber is generated. The present paper uses a vorte method to know the global characteristics of flow behavior; the effect of shape of the nozzle and the generation of pressure drop in the cushion chamber. The vorte method is powerful to know transient high Reynolds number flows unbounded flow region, because the boundary condition at infinity is automatically satisfied. In the present problem the pressure distribution is main purpose, however, it is not so easy to know the pressure distribution in two-dimensional flow solved by vorte methods. There are mainly two methods to know the pressure distribution; to use the unsteady Bernoulli equation and to solve the Poisson equation with respect to pressure. For the unsteady Bernoulli equation, there is a question: Is it applicable for high Reynolds number flows, that is, for the viscous flow? To solve the Poisson equation is reasonable, however, we have to solve numerically this equation from the vorticity field obtained by the vorte method. The present paper first shows the general approach to obtain the pressure distribution for the vorticity field given by vorte blobs with Gaussian cut-off function. Further, we discuss the relation between the unsteady Bernoulli equation and the present one. Second a method combined with the panel method on conformal mapping plane is proposed in order to save the numerical time for solving flows issued from various shapes of nozzle. Finally numerical results of the pressure distribution on the ground are shown. PRESSURE FIELD The flow treated in the present paper is assumed to be incompressible and two-dimensional Newtonian fluid flow. The governing equations are governed by the Navier-Stokes equations: u t +( u ) u = ρ p + ν 2 u, () u =, (2) where u is velocity field, p is pressure, ρ is density and ν is kinematic viscosity. From these equations () and (2), we easily have H = u ω + ν 2 u u t, (3) 2 H = ( u ω), (4) where ω is vorticity vector and H is given by H = p/ρ + u 2 /2. Here, let us consider the case where the physical plane z = + iy is mapped conformally to ζ = q + iq 2. The mapping function is given by z = f(ζ). Then, we have the following relations: 2 = h 2 2 q, (5) ( U = h 2 U + ), (6) h q 2 h where U =(U,U 2 ), h = dζ/dz and q = ē + ē 2 q 2 where ( ē, ē 2 ) is the unit vector in (q,q 2 ) plane. Taking U = u ω, we have from Eq.(4), since ω =(,,ω), 2 q H = U + h q 2. (7) h From Green theorem, we have H = H o H 2π Q N ln Q Q H N ln Q Q ds, (8) H o = ( U + ) 2π Q q h q 2 h ln Q Q dq dq 2, (9) where Q =(q,q 2 ), N and s are the normal and tangential direction of the surface of the body Q in ζ plane, respectively. From Eq.(3), we have the boundary condition of H: H N = H h n = u s h ω + ν ω s u n h t, () where n is the normal direction of the surface of the body in the physical plane and u s and u n are the velocity components on the surface in the physical plane with respect to tangential and normal directions. pproaching Q Q o on Q, we have a boundary integral equation for H. This equation has been solved numerically by Shintani, et al. (994) and Nakanishi, et al. (992). Here, we compose H as H = H o + H : 2 q H o = U +, h q 2 h () 2 qh =, (2) where H o is the particular solution: H o = H o us 2π Q h ω + ν ω s h u n t 2 Copyright c 999 by SME

3 ln Q Q ds. (3) Then, the boundary condition of H is given by H / N = H o / N. (4) where = (, y), (, 2 ) and (u,u 2 ) are the position vector and the velocity vector of the -th vorte blob, respectively, and ( ) denotes the value at =. Furthermore, we have near the -th vorte blob; z z + δ oˆq o ep(iφ)(dz/dζ), where δ o. Substituting Eqs.(2) and (2) into Eq.(9), we finally have From the definition of U, we have U /h =ũ ω, U 2 /h = ũ 2 ω, (5) H o φ Γ ep r2 t δ 2 + H o, (22) where ũ and ũ 2 are defined by ũ = u q 2 + u y q 2, ũ 2 = u + u y. (6) Here, we assume that the vorticity field is approimately epressed as follows; ω( ) = Γ ϕ, (7) δ where ( φ = Real i ) 2π ln(ζ ζ ), (23) ( H o = Γ 2 q 2 + q 2 4π 2 δ 2 E 2qq cos(θ θ ) δ 2 dζ/dz 2 ( +E 2 q2 + q 2 2qq ) cos(θ θ ) δ 2 dζ/dz 2, (24) ) where ϕ is the cut-off function and δ is the cut-off radius. In the present paper, the cut-off function is assumed to be Gaussian: ϕ() = ep( 2 )/(δ 2 π). Then, Ho defined by Eq.(9) becomes as, by integration by parts where E () = ep( 2 )d. (25) where I is defined by H o = 2π Γ I, (8) I = ϕ Q δ Q Q ũ (q 2 q ) ũ 2 (q 2 q 2 ) dq dq 2. (9) Here, the cut-off radius δ is very small, so that the velocity field (ũ, ũ 2 ) near is approimately given as ũ ( ) ũ + Γ 2π 2 ep ( 2 ) δ 2 ( y + 2 ) +( ),(2) q 2 q 2 ũ 2 ( ) ũ 2 + Γ 2π 2 ep ( 2 ) δ 2 ( y + 2 ) +( ),(2) Let us consider two typical cases: () The physical plane is mapped to the outside of the circular cylinder with unit radius in ζ plane and (2) it is mapped to the half upper plane. For case (), we can easily obtain H : where Thus, we have H = 2π 2π 2π 2π m(θ)ln ζ ep(iθ) dθ m(θ)dθ ln ζ, (26) m(θ) = 2 H o / N, for ζ = ep(iθ). (27) H = π 2π ζ cos(θ θ o ) H o dθ, (28) ζ ep(iθ) 2 where ζ = ζ ep(iθ o ). This result is the same as one given by Kida, et al. (997). We note that the accuracy of second 3 Copyright c 999 by SME

4 derivatives of φ is not good as already shown by Beale & Mada (982), so Eq.(28) is superior to Eq.(26). For case (2), we define Ĥo as Ĥ o = { H o(q,q 2 ) for q 2 > H o (q, q 2 ) for q 2 <. (29) Then, we have Ĥo/ N =. Thus, we can easily have H =. (3) Let us consider the case of u = on the surface of the body. Then, we have the boundary condition: H/ n = O(ν) on the surface of the body. The velocity field, u p, due to vorte blobs is given by u p = Γ gradφ ep r2 δ 2, (3) where r =. Here, we define the velocity potential Φ : 2 Φ =, Φ n = φ Γ ep r2 n δ 2 on Q. (32) Then, u p + gradφ is the velocity field of the physical state for high Reynolds number flow. Here, we define H as H = Φ / t + H. (33) Then, we easily have from Eqs.(2) and (4): 2 H =, (34) H n = 2 φ δ 2 Γ n n (r2 ) ep r2 δ 2 n H o + O(ν) on Q. (35) Thus, we see that H becomes very small for r δ. The thickness of the boundary layer is of the order of /Re /2 and the typical length with respect to the viscous diffusion within time step t is of the order of (ν t) /2, that is, ( t/r e ) /2. Since /Re /2 δ ( t/r e ) /2, we easily see that H is very small for high Reynolds number flows ecept neigbouhood of vorte blobs. This result implies that the Bernoulli equation, H = Φ / t+ H o, is reasonable for high Reynolds number flows. IMPINGING JETS There are mainly two typical cases on impinging et used for the floater; single et issued from a single slot and two ets or peripheral et issued from a slit whose aspect ratio is large. The former aims to promote the heat echange and the latter aims to support the strip strongly by air cushion. In the latter case, the impinging et bifurcates at the impinging point on the surface of the strip to atmospheric side and cushion side. The flow to the cushion side generates a circulatory flow which induces pressure drop, that is, this circulatory flow causes the decrease of the supporting force. To avoid this pressure drop, the nozzle is pushed out to the cushion chamber, as shown in Fig.. Figure shows the physical state near the nozzle eit. The nozzle length of the cushion side and the atmospheric side is DN and EN 2, respectively, and the angle is θ and θ 2, respectively. CE is the length of the nozzle from the outer edge of the floater to get the space for cooling duct. The lengths and the velocities are normalized with the gap height of the floater, BB, and the eit velocity at DE, respectively. Here, we assume that the aspect ratio L/B, where the width of the floater is L and its span length is B, is much large and L and B is much larger than the eit length of the nozzle t and the height of the strip. Then, when we focus on the flow near the nozzle, the flow is two-dimensional and the cushion side is unbounded, as shown in Fig.. NUMERICL METHOD To treat the various shapes of the nozzle we use the conformal mapping method together with the panel method. The basic configuration of the floater CEDBB in Fig. into ζ upper half plane as shown in Fig. 2. The mapping function is given by z = π ( ( ζ 2ζ /2 /2 )) +ln. (36) ζ /2 + Then, the nozzle, DN and EN 2, are shown in Fig.2. The configuration of the nozzle in ζ plane is decided from Eq.(36). The eit flow from the nozzle is simulated by the distribution of source, m, on DE. Here, we consider the antisymmetric vorticity field in lower half ζ plane with respect to the q ais. Then, we have m = 2 π q /2 q v o, (37) where v o is the velocity distribution of the velocity on DE in the physical plane (in the present calculation v o = ). 4 Copyright c 999 by SME

5 q 2 C N 2 E D N B q B D θ θ2 E C Figure 2. Mapping plane N β N2 B O y C p 2 5 Figure. Physical model near the nozzle eit 5 Here, we assume that the boundary layer on the surface of the strip is neglected: The Reynolds number of the impinging et based on the eit velocity at the nozzle and the width of the nozzle eit is very high and we focus on the flow near the impinging point so that the development of the boundary layer on the surface of the strip is very small. In the present paper, the shape of the nozzle DN and EN 2, is simulated by the point vorte distribution, in actual calculation, four vorte points. The et sheet is simulated by vorte points with Chorin s cut-off function, which shed from the edges of the nozzle and the strength of the shedding vorte points is constant. In the actual calculation, the strength of the bound vorte on the surface of the nozzle is obtained from the tangency condition, that is, the normal component of the velocity on the surface of the nozzle must be zero. The viscous effect is simulated by using the random walk method proposed by Chorin (973). The time marching is the first order Euler method. The relation between the physical z plane and ζ plane is obtained by iterative method. NUMERICL RESULTS The time step is.2, Reynolds number is and the eit velocity on DE is in the present calculation. The width of the nozzle N N 2 is.34, the length of the nozzle EN 2 is.57 and the height BB is. Here, C p is defined as C p =(p p )/ ( 2 o) ρv2.the pressure distribution is obtained by using the method in case(2), Eq.(3). Figure 3 is the average value of the pressure distri Figure 3. Pressure distribution on the surface of the strip in the case of θ = θ 2 =6, β =, CE = : Solid line; t =, dash line; t =2, small dash line; t =3, dotted line; t =4, chain line; t =5 bution on the surface of the strip. In this figure, the average value of before and after total 5 steps is shown. We see that the negative pressure region at the atmospheric side moves to the downward with development of time and in the cushion side the drop of the pressure appears and moves to the center. Figure 4 shows the effect of the position of the nozzle. In this figure, the average value of before and after total 5 steps is shown and we see that the cushion pressure increases a little due to the increase of CE. The pressure of the atmospheric side does not decrease so much. Figure 5 shows the effect of the angle of the nozzle eit β. We see that the cushion pressure increases with the increase of β because the curvature of et becomes large with the increase of β. Figure 6 shows the velocity distribution at the nozzle eit u. n is the distance from the edge of the atmospheric side of the nozzle to the edge of the cushion side normalized with the length N N 2. We see that the velocity at the nozzle eit is almost potential flow, that is, it is inverse proportional to n + a, where a is a constant. 5 Copyright c 999 by SME

6 25.8 C p 2 u n Figure 4. Pressure distribution on the surface of the strip in the case of θ = θ 2 =6, β =, t =5: Solid line; CE =, dash line; CE =.5, small dash line; CE =, chain line; CE =.5 Figure 6. Velocity distribution at nozzle eit in the case of θ = θ 2 =6, β =, t =5: Solid line; CE =, dash line; CE =.5, small dash line; CE =, chain line; CE =.5 Cp shapes of the nozzle. The present numerical results say that the configuration of the nozzle treated here does not generate so much large pressure drop at the atmospheric and cushion sides Figure 5. Pressure distribution on the surface of the strip in the case of θ = θ 2 = 6, t =5, CE = : Solid line; β =45, dash line; β =3, short dashed line; β =5, dotted line; β = CONCLUSIONS The present paper shows the general formula of the pressure distribution and discusses whether or not the unsteady Bernoulli equation is applicable to the viscous flow. The present paper says that to use the Bernoulli equation in viscous fluid flow to obtain the pressure distribution is available for high Reynolds number flow. The present paper shows an approach of the combination of the conformal mapping function with the panel method. This approach is shown to be very useful in order to calculate the various REFERENCES Shintani, M., Shiraishi, H. and kamatsu, T., Investigation of two-dimensional discrete vorte method with viscous diffusion model, Trans. JSME, B, 6-572, 994, -7. Chorin,.J., Numerical study of slightly viscous flow, J. Fluid Mech., 57, 973, Kida, T., Sakate, H. and Nakaima, T., Pressure distribution obtained using two-dimensional vorte method, Trans. JSME, B, 63-66, 997, Nakanishi, Y. and Kamemoto, K., Procedure to estimate unsteady pressure distribution for vorte method, Trans. JSME, B, , 992, Beale, J.T. and Mada,, Vorte method II: Higher order accuracy in two and three dimensions, Math. Comp., 39, 982, Copyright c 999 by SME

UNSTEADY LOW REYNOLDS NUMBER FLOW PAST TWO ROTATING CIRCULAR CYLINDERS BY A VORTEX METHOD

Proceedings of the 3rd ASME/JSME Joint Fluids Engineering Conference Jul 8-23, 999, San Francisco, California FEDSM99-8 UNSTEADY LOW REYNOLDS NUMBER FLOW PAST TWO ROTATING CIRCULAR CYLINDERS BY A VORTEX