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Dr Achhaibar Singh1, Dr Nickolay Zosimovych2

1Amity University, Uttar Pradesh, India

2University of South Wales, Pontypridd, UK

LAMINAR FLOW INVESTIGATION BETWEEN A STATIONARY AND A ROTATING DISK WITH A SOURCE AT CENTRE

 

The present study predicts the flow field and the pressure distribution for a laminar flow in the gap between a stationary and a rotating disk. The fluid enters the gap between two concentric disks at the centre of the disks and diverges to the outer periphery. The closed form solutions for radial velocity, tangential velocity and pressure are obtained by simplifying the Navier Stokes equations. The velocity and pressure distributions are compared with the experimental data of other investigators. The theoretical and the experimental results are found in good agreement. The present solutions deviate from the experimental results at high rotational Reynolds number. The flow separation has been predicted near the stationery disk due to the effect of centrifugal force.

Keywords: Disks; Inertia; Laminar; Viscous.

 

I. Introduction. The flow between two disks is a topic of much interest to mathematicians as well as to engineers. This problem allows a similarity solution with a potential of a multiplicity of the solutions, a matter of great interest to theoreticians. The present flow finds several applications such as face seals, disk type heat exchangers, centrifugal manometers, turbo machines and lubrication. Therefore, it is equally interesting to the engineers.

           Lee and Lin [1] simplified the Navier Stokes equations and obtained expressions for pressure distribution for the inward flow between two stationary disks. They used a linearization technique to obtain pressure without deriving expression for velocity. Wang et al. [2] investigated the flow between shear-force pump with multiple corotating disks numerically as well as experimentally. The significant friction loss was caused by the long flow path due to strong recirculation in the diffuser and scroll volute, which was found in the simulation results for the internal flow in the whole pump system. In addition, a reverse flow appeared in the rotor part at a low flow coefficient, which significantly deteriorated the rotor performance. Biswas et al. [3] carried out numerical investigation of outward flow between two stationary disks. The flow field consists of several features like toroidal recirculation, annular separation bubble and flow reattachment. Numerical investigation was performed by Al-Shannag et al. [4] on isothermal laminar flow of air between two corotating confined in a fixed cylindrical enclosure. There is flow exchange from the disks space and housing space through the clearance. They showed that increasing gap size decreases disk surface shear and the associated disk torque coefficient, but at the cost of destabilizing the inter-disk flow.

The outward flow between two stationary disks has been of much interest in connection with lubrication technology where viscous forces were predominant as compared to inertia forces. McGinn [5] employed dye injection technique to investigate streamline configuration, flow separation, and cavitation. Several investigators carried out theoretical study using Karman’s momentum integral method and power series method for the outward laminar flow [6-7]. Moller [8] investigated laminar and turbulent flows experimentally and theoretically. He concluded that the viscous sublayer thickening was the cause of relaminarization of the flow.

Aim of the present study is to derive the closed form solutions for velocity and pressure distributions by simplifying the Navier Stokes equations. The closed form solutions are compared with the published experimental data of other investigators. The parametric study has been carried out to understand the effect of the through flow Reynolds number, the rotational Reynolds number and the gap ratio on the outward flow between a stationary and a rotating disk.

II. Mathematical Model. The geometry and the coordinate system are shown in Fig. 1. Two parallel disks of outer radius  one stationary and the other rotating with a constant angular speed are kept h distance apart. The gap between the disks is small as compared to the outer radius. Therefore, the axial component of the flow velocity is negligibly small as compared to the radial and the tangential components. The fluid enters the radial channel through a hole at the center of the disks and diverges towards the outer periphery. The net flow in the gap between the disks is outward. The flow is axisymmetric, i.e., derivatives of the variables with respect to polar coordinate are zero.

III. Results and Discussion.

3.1. Model of a Control System for the Launch Vehicle. A control system of the launch vehicle is designed to maintain the required (programmed) trajectory parameters of the center of mass and around the center of mass (Fig. 1)

Fig. 1. Geometry and coordinate system

 

The present flow is governed by the continuity and the momentum equations in the cylindrical coordinates:

(1)

(2)

(3)

(4)

The boundary conditions are:

 at  at

(5)

In order to get a solution for the tangential component of velocity, Eq. (3) is reduced to an ordinary differential equation by introducing the Karman’s similarity form as

(6)

The substitution of the above expression for v into Eq. (3), yields

(7)

 Equation (7) is a second order ordinary differential equation that can be solved by replacing,   with   where,   as suggested by Lee and Lin [1]:

(8)

where,

Corresponding boundary conditions are

 at   at

(9)

The solution of Eq. (8) can be expressed as

(10)

Using Eqs. (6) and (10), the tangential velocity can be written as

(11)

The non-dimensional forms of Eqs. (11) is given as:

;

 

 

(12)

Where,    Here, and are through flow Reynolds number, and gap ratio,

Eq. (12) was integrated using a second order Runge-Kutta method. The truncation errors were limited to the order of 10-6 by choosing a step size of 0.01.

3.2. Simulating Velocities Depends on Through Flow Reynolds Number.

Non-dimensional tangential velocity depends on through flow Reynolds number, gap ratio and radial location (Eq. (11)). The tangential velocity at Req = 286624 is compared with the published experimental data [9] and is found reasonably in good accord with the experimental result (Fig. 2). The angular velocity of the fluid, away from the rotating disk, decreases with the increase in through flow Reynolds number. Radial convection of tangential velocity can be observed in Fig. 3 where the tangential velocity increases near the stationary disk in the direction of flow. The dominance of through flow decreases axial convection of the tangential momentum. With increase in gap ratio, the tangential velocity decreases near the stationary disk and increases near the rotating disk (Fig. 4).

Fig. 2.  Tangential velocity distribution for different Reø=50000,   g=0.03,

Fig. 3.  Tangential velocity distribution for Req=1000, g=0.03

Fig. 4.  Tangential velocity distribution for Req=1000,

               

Figure 5 presents the radial velocity profile for different through flow Reynolds numbers at a fixed rotational Reynolds number of 8000, a gap ratio of 0.02 and a radial position of 0.7. The centrifugal force imparts additional momentum to the fluid near the rotating disk. As a result, the fluid near the rotating disk moves faster than the fluid near the stationary disk. Therefore, the maximum radial velocity shifts toward the rotating disk despite the symmetric boundary condition for radial velocity.

Fig.5. Radial velocity distribution for Reø=8000, g=0.02,

 

The plots of radial velocity profile at different gap ratios, for Req=10, Reø= 1000, =0.9, show an increase in the back flow region with an increase in the gap ratio (Fig. 6).  As gap ratio increases the radial velocity becomes asymmetric and further increase in the gap ratio results in flow separation and backflow at stationary disk.

Fig. 6.  Radial velocity distribution for Req=10, Reø=1000, 

                                                                

 Figure 7 compares the experimental results [10] with the present solution for Req=41, Reø=12279, g=0.0046. It is evident from the figure that both the results are in good agreement. According to the Eq. (12), the pressure consists of the pressure drop due to viscous dissipation, convective inertia and rotational inertia. The figure indicates that the rotational inertia dominates the viscous dissipation and convective inertia resulting in negative pressure drop. As through flow Reynolds number increases the effect of rotational inertia decreases and the pressure drop becomes positive everywhere in the radial channel for Req=100.

Fig. 9.  Pressure distribution for Reø =12279, g=0.0046

 

IV. Conclusions.

The linearization has simplified the problem resulting in the derivation of the expressions for velocity and pressure. The present solution predicts velocity and pressure fields that are comparable with the experimental data.  The radial convection influences the distribution of tangential velocity between the disks.

The centrifugal force causes backflow near the stationary disk. An increase in the through flow Reynolds number decreases the backflow and an increase in the gap ratio enhances the backflow. The backflow decreases in the direction of flow due to a decrease in the centrifugal force.

The pressure drop increases with a decrease in the through flow Reynolds number as well as the gap ratio due to high viscous losses. An increased rotational Reynolds number increases the flow path resulting in high viscous losses that causes the pressure drop to increase.

 

 

 

References

1.     Lee P.M. and Lin S. Pressure distribution for radially inflow between narrowly spaced disks, Trans. ASME J. Fluids Engg., No107, pp. 338-341, 1985.

2.     Wang B., Okamoto K., Yamaguchi K., Teramoto S. Loss mechanisms in shear-force pump with multiple corotating disks, ASME J. Fluids Eng., No136, pp. 081101-1-081101-10, 2014.

3.     Biswas N., Manna N. K.,  Mukhopadhyay A.,  Sen S. Numerical simulation of laminar confined radial flow between parallel circular discs, ASME J. Fluids Eng., No134, pp. 011205-1- 011205-8, 2012.

4.     Al-Shannag M., Herrero J., Humphrey J. A. C., Giralt F. Effect of radial clearance on the flow between corotating disks in fixed cylindrical enclosures, ASME J. Fluids Eng., No124, pp. 719–727, 2002.

5.     McGinn J. H. Observations on the radial flow of water between fixed parallel plates, Applied Scientific Research, No5, pp. 255-264, 1956.

6.     Woolard H. W.  A theoretical analysis of the viscous flow in a narrowly spaced radial diffuser, ASME J. Appl. Mech., No24, pp. 9-15, 1957.

7.     Savage S. B. Laminar radial flow between parallel plates, ASME J. Appl. Mech., No31, pp. 594-596, 1964.

8.     Moller P. S. Radial flow without swirl between parallel disks, Aeronaut. Quart., No14, pp. 163-186, 1963.

9.     Bayley F. J., Owen J. M. Flow between a rotating and a stationary disk, Aeronaut. Quart., No20, pp.333-341, 1969.

10.      Szeri A. Z., Adam M. L. Laminar through flow between closely spaced rotating disks, J. Fluid Mech., No86, pp. 1-14, 1978.