Journal of Engineering Physics and Thermophysics, Vol. 85, No. 2, March, 2012

AN APPROXIMATE MODEL OF THE DYNAMICS AND HEAT TRANSFER OF AN IMPACT CYLINDRICAL IDEAL LIQUID JET B. A. Uryukov, V. D. Belik, and G. V. Tkachenko

UDC 532.5.031;536.24

A jet model based on approximations of velocities, satisfying the continuity equation, and on the integral momentum equation is presented. The solution for the jet dynamics turned out to be nonmonotonic: as an obstacle recedes over a distance larger than a certain critical one, the jet escapes from the receiver nozzle rectilinearly and remains unchanged until the distance to the obstacle becomes equal to the critical one, whereupon the jet begins to spread. The heat transfer law has been determined on the basis of the momentum and boundary layer energy equations written in an integral form. They were solved by the Squire method. It is shown that with decrease in the distance to the obstacle, if it is smaller than the critical one, the Nusselt number at the stagnation point increases. Keywords: impact jet, ideal liquid, heat exchange between a jet and a plane wall. The problem of impingement of an ideal liquid jet on a plane obstacle is of great interest for both practical applications and for theoretical hydromechanics. Despite the apparent simplicity, it has not been solved as yet in full even in the plane case. The solution is known for a plane jet issuing from infinity, which was obtained by the method of the theory of functions of a complex variable (TFCV) [1–3]. However, here one needs to carry out serious numerical calculations to obtain quantitative results. The axisymmetricity of the problem makes its solution much more difficult. Thus, in [4] a variational method based on formulation of an extreme problem is applied to determine the velocity in a jet. In the long run, the authors cite only tabulated data for the jet velocity distribution on the obstacle surface and only in one case, viz., for the distance between the nozzle cut and the wall equal to the diameter of the nozzle cut. In [5], the collision of an axisymmetric jet, flowing out an arbitrarily shaped nozzle, with a wall of arbitrary profile is considered. With the aid of the potential theory of a simple layer, the problem is reduced to the solution of an integral equation with a complex singular kernel. In essence, the solution of such an equation represents a new problem the concrete lines of attack on which have not been indicated. In [1–5], the jet flow rate is given initially and, consequently, account is not taken of the fact that the presence of an obstacle in the path of the jet influences not only the trajectories of streamlines but also the jet flow rate, since it raises the pressure at the nozzle cut. This is especially important in the technologies of surface treatment of articles during deposition of coatings on them where one tries to keep to a minimum the distance from the nozzle cut to the obstacle to decrease the losses of both the thermal and kinetic energies of the jet and particles. The model proposed in the given work for an axisymmetric jet is based on velocity approximations satisfying the continuity equation and the integral momentum equation. To analyze the heat exchange between a jet and a plane wall, the equations of momentum and of the boundary layer energy written in an integral form were used. Jet Dynamics. The schematic of the jet flow is presented in Fig. 1. An incompressible liquid flows out from the receiver A through a nozzle of circular cross section. In the receiver the liquid is at rest under the pressure p00. The jet boundary R(z) separates the moving liquid and the surrounding quiescent medium, which is under the pressure pa. It is considered that the surrounding medium does not penetrate into the jet. At the boundary, V is constant due to the constancy of pressure, and it is calculated from the Bernoulli equation: I. N. Frantsevich Institute of Materials Science Problems, National Academy of Sciences of Ukraine, 3 Krzhizhanovskii Str., Kiev, 03142, Ukraine; email: [email protected]. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 85, No. 2, pp. 299–304, March–April, 2012. Original article submitted March 17, 2011. 1062-0125/12/8502-03172012 Springer Science+Business Media, Inc.

317

Fig. 1. Schematic of the jet flowing out from the reveiver and of its collision with the obstacle. 2

V = (p00 − pa) ⁄ 2ρ .

(1)

We prescribe the velocity components vr and vz in the form of the products of functions of the coordinates r and z: vr = V sin α

f (r) F (r) , vz = V cos α . F (R) f (R)

(2)

These relations satisfy the condition of equality of the velocity components at the jet boundary to the corresponding 1 ∂rvr ∂vz + = 0, we obtain projections of the velocity V. Substituting Eqs. (2) into the continuity equation r ∂r ∂z sin α 1 d [rf (r)] d ⎡ cos α ⎤ + F (r) ⎢ =0. dr dz ⎣ F (R) ⎥⎦ f (R) r 1 d[rf(r)] 1 d[Rf(R)] and correspondingly that F(R) = . Cancelling the functions r, we r dr R dR find an equation for determining the shape of the jet boundary: From this it follows that F(r) =

sin α d ⎛ R cos α ⎞ + =0. f (R) dz ⎜⎝ [Rf (R)]′ ⎟⎠ Taking into account that tan α =

(3)

dR d dR d and performing the substitution = + , we find the first integral dz dz dz dR

of Eq. (3): dR = dz

⎯⎯⎯⎯⎯⎯ √

⎛ R2f (R) ⎞2 C⎜ ⎟ −1 . ⎝ [Rf (R)]′ ⎠

(4)

It is precisely the solution of this equation that satisfies the initial condition z = 0, R = R0 and is extended to the obstacle where z = H, R → ∞ has the form ∞

H=∫

R0

318

dR

⎯⎯⎯⎯⎯⎯ √

⎛ R2f (R) ⎞2 C⎜ ⎟ −1 ⎝ [Rf (R)]′ ⎠

(5)

which is one of the conditions for determining the relationships between the parameters of the problem. We will take the momentum equation for the given problem in an integral form having written the projections of forces onto the direction z and on the planes z = 0 and z = H: R0

R

R

R0

0

∫ (p + ρvz )z=0 rdr + ∫ pardr = ∫ pz=H rdr . 2

0

This equation can also been written in the form R0

R

∫ (p − pa + ρvz )z=0 rdr = ∫ (p − pa)z=H rdr . 2

0

0

The pressure on the planes is determined from the Bernoulli equation p = p00 − p00 −

ρν2z ρν2r − at z = 0 and p = 2 2

ρν2r at z = H. Substituting the expressions for the velocities and letting R → ∞, we obtain 2 R0

2

2

∞

2

2

2 2 ⎡ ⎛ sin α⎞ ⎤ ⎡ ⎛ 1 d [rf (r)] ⎞ ⎛ R cos α ⎞ ⎛ sin α ⎞ ⎤ ∫ ⎢1 + ⎜ r dr ⎟ ⎜ [Rf (R)]′ ⎟ − f (r) ⎜ f (R) ⎟ ⎥ rdr = ∫ ⎢1 − f (r) ⎜limz→H f (R) ⎟ ⎥ rdr . ⎠ ⎝ ⎝ ⎠z=0 ⎠z=0⎦ ⎝ ⎝ ⎠⎦ 0⎣ 0⎣

Resorting to Eq. (4), we find 2

2

1 1 ⎛ sin α ⎞ ⎛ R cos α ⎞ , = ⎟ = ⎟ ⎜ ⎜ 2 2 2 ⎝ [Rf (R)]′ ⎠z=0 CR f (R0) ⎝ f (R) ⎠z=0 f (R0) We assume that R → ∞, f(R) → 1 when z → H, then limz→H

2

⎛ 1 [R0 f (R0)]′ ⎞ ⎟. ⎜1 − 2⎟ 2 ⎜ C R0 [R0 f (R0)] ⎟ ⎜ ⎠ ⎝

sin α → 1. As a result of the substitution into the pref(R)

vious equation, we find the function C(R0): R0

C=

2 R0

2 [R0 f (R0)]′ [rf (r)]′ ∫ r dr + 2 2 R f (R )

1

0

0

2 2

R0 f (R0) ∞

∫ [1 − f 0

2

(r)] rdr +

1 2

∫f

2

(r) rdr

0 0 R0

∫f

f (R0) 0

2

(r) rdr −

2 R0

.

(6)

2

Further solution of the problem is connected with the concretization of the function f(r). The conditions that should be satisfied by this function are determined by the physically obvious behavior of velocities. First, f(r) must depend linearly on r in the vicinity of the jet axis. Second, its second derivative at the point r = 0 must be equal to zero. In this case, the distribution of the longitudinal velocity over the radius will be smooth in the vicinity of the axis. On fulfillment of this condition, the momentum equation is satisfied on the axis, which in the given case is equivalent to the absence of vorticity. Finally, in the case of an infinite increase in the argument, the function f(r) must tend to unity, which corresponds to the equalization of the radial velocity on the obstacle far from the point of stagnation with the boundary velocity V. r We introduce the dimensionless radial coordinate η = β in which β is the form parameter unknown beforeH hand. Relations (4)–(6) also allow us to find the constant C and the shape of the boundary, i.e., to solve the problem completely. We investigated two kinds of the function f(η) that satisfy the conditions posed;

319

Fig. 2. Dependence of kG (1) and R′0 (2) on the ratio H ⁄ R0.

1 2 f (η) = 1 − exp ⎛⎜− η − η ⎞⎟ , 2 ⎠ ⎝ f (η) =

(7) (8)

m 2 arctan (η + aη ) . π

In Eq. (8) the coefficient a can have an arbitrary value for m > 2 (if m = 2, a = 0). Also, the function f(R) has a corresponding form. The form of Eq. (8) is convenient because by equalizing the parameters a and m it can be selected in such a way as to best correspond to the exact solution. As was already said above, one of the goals of the present problem is the determination of the influence of an obstacle on the jet flow rate. Therefore we introduce into consideration the coefficient of discharge kG which is the ratio of the flow rate G under the given conditions to the flow rate G0 with the velocity of jet outflow from the nozR

zle equal to V, i.e., G0 =

ρVπR20.

The jet flow rate in any trasverse cross section is equal to G = 2πρ∫vzrdr. Substi0

tuting the expression for vz into the latter expression, we obtain 1 d [rf (r)] r dr G = 2πρV ∫ cos α rdr . 1 d [Rf (R)] 0 R dR R

Integrating and using relation (4), we find kG =

2 2 ⎯⎯C R0√

.

(9)

The results of calculation of the parameters R′0 and kG for (7) are given in Fig. 2 where it is seen that the solution (R′0 ≥ 0) exists only if the parameter H ⁄ R0 is smaller than a certain critical value. The critical point is characterized by the following values: (H ⁄ R0)cr = 1.002 and kGcr = 1.227. As would be expected, in the range of the solution existence, with decrease in the distance from the nozzle cut to the obstacle, the angle of the jet turning increases and the discharge coefficient decreases. The only physical interpretation of the monotonic solution obtained is that, with the obstacle being removed a distance larger than the critical one, the jet will outflow from the nozzle rectilinearly (R′0  0) and will remain so until the distance to the obstacle becomes equal to Hcr, after which it begins to 320

Fig. 3. Dependence of vr(H) ⁄ V on r ⁄ R: 1) numerical solution of [4]; 2) approximate solution for (7); 3) approximate solution for (8). Fig. 4. Dependence of R ⁄ R0 on z ⁄ R0: 1) numerical solution of [4]; 2) approximate solution for (8). spread. In this case, the flow rate remains constant (the dashed line in Fig. 2). The substitution of Eq. (8) did not alter the result. The solutions obtained were compared with the numerical data of [4] on the distribution of the jet radial velocity over an obstacle. In the present work, the velocity distribution is given for the obviously supercritical distance H ⁄ R0 = 2 (both for (7) and (8), in the range of the velocity approximation parameters 0.1 ≤ a ≤ 10; 3 ≤ m ≤ 7, the critical value of H ⁄ R0 lies near unity). Consequently, the data of [4] can be compared with the approximate solutions for the critical distance H ⁄ R0. The results of comparison of the numerical and approximate solutions of the radial velocity distribution of the jet over an obstacle are given in Fig. 3 where it is seen that the form of (7) differs substantially from the numerical solution. For (8) we managed to select such values of the numbers a and m (a = 0.3, m = 4.6) that provide good agreement with the exact solution in the most interesting zone, namely, directly under the nozzle cut, i.e., at r ⁄ R0 ≤ 1. The critical distance to the obstacle in this case is equal to (H ⁄ R0)cr = 1.066 and the limiting coefficient of jet discharge is equal to kGcr = 1.015. It should be noted that the limiting coefficient of the jet discharge must be equal to unity, since, as follows from the given solution, the distance from the nozzle cut to the obstacle exceeding the critical one does not influence the conditions of jet outflowing from the receiver. Therefore in selecting the values of a and m, it is necessary that, apart from the correspondence of the velocities, one should see to it that the value of kGcr is as close to unity as possible. The differences in the description of the jet boundary when the distance between the nozzle and the obstacle is larger than the critical one are shown in Fig. 4. Heat Exchange with the Obstacle. The dynamic boundary layer in an axisymmetric flow on a plane wall is described by the momentum equation in an integral form: 1 d

2

[rVr δ2] + δ1

1 dVr

=

cf

=

τw

.

(10)

1 d qw [rVrδth1] = St = . rVr dr ρVrc (Tw − T∞)

(11)

2

rVr dr

Vr dr

2

2

ρVr

The thermal boundary layer is described by the energy equation:

If we approximate the velocity and temperature by a certain function, the solution for dynamic and thermal boundary layers can be found by a method similar to Squire’s for a plane plate. To simplify the solution, we resort to an approximate solution for the ratio of the thicknesses of a thermal and dynamic boundary layers [6], i.e., Δ = δth  Pr−1 ⁄ 3, and we assume self-similarity of the velocity and temperature profiles: δ 321

Fig. 5. Dependence of Nu ⁄ Nu0 on r ⁄ R0 at H ⁄ R0 = 1.066 (1), 0.533 (2). Fig. 6. The Nusselt number at the stagnation point vs. the parameter H ⁄ R0. vr Tw − T ⎛z⎞ ⎛ z ⎞ =Φ⎜ ⎟, =Φ⎜ ⎟. Vr ⎝ δ ⎠ Tw − T∞ ⎝ δth⎠ It is assumed in Squire’s method that Φ(x) = 2x − 2x3 + x4. Taking into account that τw = μ

∂vr ⎪ and sub⎪ ∂z ⎪z=0

stituting the approximation of the velocity profile into the definitions of the boundary layer thicknesses, we find the solution of Eq. (10): r 2

δ =

2μΦ′ (0) ρΔ2

2

3+2b

2

4+2b

∫ r Vr

dr

0

,

r Vr

where b = Δ1 ⁄ Δ2, Δ1 = δ1 ⁄ δ = 0.3, Δ2 = δ2 ⁄ δ = 0.1175. Using the definition of the heat flux to the wall qw and the Nusselt number qw = −

qwd0 ∂T ⎪ , for Eq. (8) we find ⎪ , Nu = λ(Tw − T∞) ∂z ⎪z=0 Nu = Nu0

2+b

ηf (η)

(12)

.

⎯⎯⎯⎯⎯⎯⎯ √ η

2 3+2b 2 (6 + 2b) ∫ η f (η) dη π 0

The Nusselt number at the stagnation point is 1⁄3

Nu0 = Pr

1⁄2

Re

⎯⎯⎯⎯⎯⎯⎯ √ 2

π

F ′ (0) (6Δ2 + 2Δ1) θ0 , θ0 = β

R0 . H

(13)

The Reynolds number is determined in terms of the nozzle section diameter and the velocity V on the jet boundary. Figure 5 shows Eq. (12) for two values of the distance between the nozzle and obstacle: the critical one (curve 1) and that twice as small (curve 2). The characteristic feature of this distribution is that within the projection of the nozzle section onto the obstacle (r ⁄ R0 ≤ 1) the Nusselt number is pr actically constant. This is due to the fact that her e the r adial velocity gr adient is also pr actically constant (see Fig. 3). As H decr eases, the fall in the Nusselt number in the r egion with r > R0 becomes sharper, but the Nusselt number at the stagnation point increases, as is shown in Fig. 6.

322

In conformity with the characteristic feature of the formation of the jet boundary described above, we may conclude that at distances between the nozzle and obstacle larger than the critical one, the conditions of heat exchange between the jet and the wall do not depend on H.

NOTATION a and m, parameters of velocity approximation; b, ratio of the thickness of displacement and momentum loss; c, heat capacity, J ⁄ (kg⋅K); C, integration constant; cf, coefficient of friction; d0, nozzle diameter, m; F, function of radial coordinate in the approximation of longitudinal velocity; f(r), f(R), and f(η), functions in velocity approximations; G, liquid flow rate in the jet, kg ⁄ s; G0, jet flow rate in the absence of an obstacle, kg ⁄ s; H, distance from the nozzle to the obstacle, m; kG, coefficient of liquid discharge in a jet; Nu, Re, Pr, and St, the Nusselt, Reynolds, Prandtl, and Stanton numbers; Nu0, the Nusselt number at the stagnation point; p, pa, and p00, pressure in the jet, surrounding medium, and in the receiver, respectively, Pa; qw, heat flux directed into the wall, W ⁄ m2; R(z), jet radius, m; R0, initial jet radius (nozzle radius), m; r and z, cylindrical coordinates, m; T, T∞, and Tw, temperature in the boundary layer, at the boundary layer edge, and of the wall, respectively, K; V, liquid velocity at the jet boundary, m ⁄ s; Vr, radial velocity of the liquid on the obstacle in a potential flow, m ⁄ s; vr and vz, radial and longitudinal velocities of liquid in the jet, m ⁄ s; x, arbitrary argument in the function Φ(x); α, angle of jet boundary inclination toward the axis; β, form parameter; δ and δth, thicknesses of the dynamic and thermal boundary layer, m; δ1 and δ2, thicknesses of displacement and momentum loss, m; δth1, energy loss thickness, m; Δ, ratio of the thicknesses of thermal and dynamic boundary layers; Δ1, ratio of the thickness of displacement to the boundary layer thickness; Δ2, ratio of the momentum loss thickness to the boundary layer thickness; η, dimensionless radial coordinate; θ0, parameter in Eq. (13); λ, thermal conductivity, W ⁄ (m⋅K); μ, viscosity, kg ⁄ (m⋅s); ρ, liquid density, kg ⁄ m3; τw, friction stress on the wall, Pa; Φ, approximation of velocities and temperature of the liquid in the boundary layer. Indices: 00, receiver; 1, 2, number of the boundary layer form parameter; a, surrounding medium; cr, critical parameters; f, friction; th, thermal; w, obstacle; ∞, boundary layer edge; ′ , differentiation with respect to the argument.

REFERENCES 1. 2. 3. 4. 5. 6.

M. I. Gurevich, Theory of Ideal Liquid Jets [in Russian], Nauka, Moscow (1979). M. A. Lavrent’ev and B. V. Shabat, Methods of the Theory of Functions of a Complex Variable [in Russian], Nauka, Moscow (1965). L. R. Loitsyanskii, Mechanics of Liquids and Gases [in Russian], Nauka, Moscow (1978). P. G. Shtern, E. A. Rudenchik, and A. K. Kerimov, Axisymmetric potential flow originating on direct impact of a jet upon a plane wall, Dokl. Akad. Nauk SSSR, 317, No. 5, 1085–1088 (1991). A. P. Chernyaev, A problem of an axisymmetric jet flow, Dokl. Akad. Nauk SSSR, 317, No. 5, 1070–1075 (1991). H. Schlichting, Boundary Layer Theory [Russian translation], Nauka, Moscow (1977).

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