Effect of Heat Transfer on Boundary Layer Stability and Transition
EBRU SARIGÖL*, KERİM YAPICI+, SENEM ATALAYER*,
SERKAN ÖZGEN*
* Department of Aerospace Engineering
+ Department of Chemical Engineering
Middle EastTechnicalUniversity
06531, Ankara
TURKEY
Abstract: - The effect of heat transfer on the boundary layer stability on a flat plate when the flow is subjected to heating from the wall has been investigated. By the use of linear stability theory, critical and transition Reynolds numbers has been calculated for different fluids (air and water). For the mean flow, momentum and energy equations on the flat plate are used whereas the stability equations which are derived starting from two-dimensional Navier-Stokes equations are used for the perturbations. The effect of temperature difference on critical and transitional Reynolds number is analyzed and compared with the analytical, numerical and experimental data in the literature.
Key Words: - heat transfer, boundary layer stability, laminar-turbulent transition, variable properties
1 Introduction
Most of the previous studies that had been done to understand the transition from laminar flow to turbulent flow have been focused on the problem why the flow does not remain laminar instead of turbulence. Most of the researchers made analyses by using the linear stability theory which has proved itself, repeating the results of experimental studies of Schubauer and Skramstad [1] for different geometries and flow regimes.
The importance of the subject that is analyzed in this present study comes from the direct relevance with the practical applications such as heated deicing equipments on aircraft and heat exchangers.
The problem of boundary layer transition has been the subject of research projects sponsored by NACA in the late 40s [1,2]. For instance, Liepmann and Fila [2] have investigated the flow on a heated flat plate and the effect of temperature difference on stability and transition and they have found that increasing the surface temperature results in the increased instability and early transition.
The study of Hauptmann [3] is worthwhile due to the reason that he found an analytical relationship between the heat transfer velocity and instability point. By using perturbation method, Hauptmann, who investigated low and moderate heat transfer rates, found that increase in the surface temperature stabilizes the liquid flow whereas the air flow is destabilized.
Wazzan, Okamura and Smith [4] have investigated a similar situation in their numerical study. They have done a linear stability analysis of water over a flat plate for different temperature differences by solving an extended version of the Orr-Sommerfeld equation. They have determined that increasing surface temperature stabilizes water flow in general and for different surface temperature values they found the critical Reynolds numbers (based on displacement thickness) varying in the range between 520 and 16000.
Herwig and Schäfer [5] have worked on a general method that examines the effect oflow heat transfer rates on the stability characteristics. Results of this analysishold for all Newtonian fluids instead of just one particular fluid. The variables are expressedinTaylor seriesexpansions with respect to temperature and pressure. The stability analysis has been done by the solution of extended version of Orr-Sommerfeld equation. Decreasing the viscosity in the near-wall regions destabilizes the flow whereas the gradual decrease of viscosity from near-wall region to free streamstabilizes the flow.
In a more present study, Schäfer, Severin and Herwig [6] examined the effect of variable fluid properties on stability by solving the extended version of the Orr-Sommerfeld equation which have been also cited in [5]. The asymptotic solutions obtained are valid for all Newtonian fluids with the Prandtl number as the only parameter left.
In the present study, the stability and transition problems of air and water over a flat plate have been investigated by using the linear stability theory. Smith-Van Ingen en method is employed for the transition calculations. The effect of temperature difference on critical and transitional Reynolds numbers is examined for both fluids and they are compared with the analytical, numerical and experimental data in literature.
2 Theory and Mathematical Formulation
The equations of motions are two-dimensional Navier-Stokes equations with variable density and viscosity for incompressible flow.
The velocity components and pressure are decomposed into a steady mean value and unsteady part:
(1)
As the mean flow satisfies the equations of motion, the corresponding mean flow terms in the resulting equations are subtracted out. The quadratic terms are also dropped. And finally, parallel flow assumption is applied to the system. For the disturbances, normal mode analysis is employed. According to this:
(2)
Here, Ψ defines any fluctuating component with defining its amplitude. The complex frequency is defined as where the real part is the wave frequency and the imaginary part is the amplification rate. The wave number whereas, is represented by α. Substituting these into the equations of motion and making necessary rearrangements, one can get a system of dimensionless equations as:
(3)
(4)
(5)
where (') denotes the derivatives with respect to the dimensionless normal distance y. In these equations, nondimensionalization is made with respect to , , and where the subscript ‘e’ refers to the values at the edge of the boundary layer. The pressure and the viscosity are nondimensionalized with respect to and , respectively. Corresponding boundary conditions are as follows:
(6)
This set of equations can be rewritten as a system of first order ordinary differential equations by defining the following new variables:
(7)
And the new system obtained will be as follows:
(8)
(9)
(10)
(11)
The boundary conditions are given as follows,
as y (12)
3 Solution Method
In the free stream, which is far away from the wall the mean flow is uniform and all gradients of y assumed to be zero. For this respect, equations (8)-(11) have constant coefficients and as such,they allow solutions of the form:
(13)
where are the characteristic values, and are the four component solution vector and the characteristic vector corresponding to , respectively.
Homogeneous equations with homogeneous boundary conditions have been obtained. Therefore such system can be solved as a characteristic value problem. Characteristic values of this system:
(14)
Because of the free stream boundary conditions (equation (12)) only the characteristic values with a negative sign are relevant. The solution corresponding to is the inviscid solution whereas the solution corresponding to is the viscous solution. General solution can be found as follows:
(15)
This solution (equation (15)) provides the initial conditions for the integration of equations (8)-(11). A variable step size fourth order Runge-Kutta method has been used as integration proceeds from the free stream towards the wall. Gram Schmidt orthonormalization technique has been used at regular intervals to maintain the linear independency of the solutions. When the wall is reached, the wall boundary conditions (equation (12)) must be satisfied. To obtain the stability diagram, two variable Newton iteration method has been used.
The mean velocity and its gradients in the above equations must be calculated with reasonable accuracy. Moreover, temperature dependent viscosity and density require the solution of the temperature field as well.
Modified Blasius equations must be solved for the velocity field:
(16)
where . As the viscosity and density are dependent on the temperature, the energy equation has to be solved as well:
(17)
where is the dimensionless temperature.Prandtl number is assumed to be 0.72 for air until 300 K. Sutherland and ideal gas equation has been used for variation of viscosity coefficient and density respectively.
(18)
Gaster’s transformation has been used to convert the results of the temporal amplification problem and transition prediction tool is the en method which is a semi empirical method based on linear amplification of unstable modes found by Smith and Van Ingen.
As far aswater is concerned, the system of stability equations does not change but proper variations of density, viscosity and Prandtl number with temperature have to be accountedfor. These are given by the empirical relations presented below:
(19)
(20)
(21)
All variables are in SI units. Here T*is temperature in Kelvin and Tris the dimensionless temperature defined as . The variation of density was not neglected although it is very small.
4 Results and Discussions
As it can be seen from Fig. 1, the interval between unstable wave numbers increases with increasing surface temperature which also results in the decrease of critical Reynolds number. This consequence can better be seen in Fig. 2. At high temperature differences, neutral stability curves are very similar to those of inviscid stability curves.
Fig.1 Neutral stability curves for various temperature differences for air.
Fig.2Variation of critical Reynolds number with temperature difference for air.
The reason of the situations observed is the decrease of air viscosity with increasing temperature. Due to this reason, there results a point on the velocity profile at which the second derivative is equal to zero (inflection point) and this plays an additional role in the instability according to Rayleigh’s theorem. The numerical results obtained are in fairly good agreement with data available in literature when compared (Fig. 3, 4 and 5).
The situation is rather interesting for water. For small temperature differences, as the wall temperature increases the flow stabilizes, but beyond 40oC the situation reverses (Fig. 6, 7 and 8). This consequence is related with the change of viscosity with Reynolds number; i.e. viscosity has a stabilizing effect for low Reynolds number, but on the contrary it has a destabilizing effect for high Reynolds numbers. When the results are compared with the data in literature, they are in good agreement (Fig. 8, 9 and 10). The numerical results of this study agree with those of Wazzan, Okamura and Smith [4]. The authors of this study think that the quantitative difference between these results is due to the different stability equations solved. Moreover, the authors of the paper mentioned above did not comment on how the variation of Prandtl number was treated.
Fig.3 Comparison of critical Reynolds number with literature data for air.
Fig.4 Comparison of computed critical Reynolds number with literature data for air.
Fig.5 Comparison of computed critical Reynolds number with literature data for air.
Fig.6. Neutral stability curves for various temperature differences for water.
Fig.7 Neutral stability curves for various temperature differences for water.
Fig.8 Variation of critical Reynolds number with temperature difference for water.
Fig.9 Comparison of computed critical Reynolds number with literature data for water.
Fig.10Comparison of transition Reynolds number with literature for water.
5 Conclusion
In this study, the effect of temperature difference on the flow over a flat plate for different fluids, namely air and water, is investigated when Mach number is equal to zero. The approach and the solution method which applies to a wide range of temperature differences,make this study more general because the solution methods in the literature are applicable only for low or moderate temperature differences. The obtained results are compared with the data in literature and it is observed that they are in good agreement qualitatively and quantitiavely. In the case of air flow, both critical and transition Reynolds numbers decrease with the increasing temperature difference between the surface and free stream and the flow is destabilized. When the fluid is water, the situation is the opposite. First of all, increasing the temperature difference stabilizes the flow and hence, increases both critical and transition Reynolds numbers, but after a certain temperature value increasing the temperature difference more begins to destabilize the flow and hence decrease the Reynolds numbers.
References
[1] Schubauer G. B. and Skramstad, H. K., Laminar boundary layer oscillations and transition on a flat plate, NACA Report No. 909, 1948.
[2] Liepmann, H. and Fila, G. H., Investigations of effects of surface temperature and single roughness elements on boundary layer transition, NACA Report No. 890, 1947, pp.587-597.
[3] Hauptmann, E. G., The influence of temperature dependent viscosity on laminar boundary layer stability, Int. J. Heat Mass Transfer, Vol. 11, 1967, pp.1049-1052.
[4]Wazzan, A.R., Okamura, T. and Smith, A.M.O., The stability of water flow over heated and cooled flat plates, J. Heat Transfer, Vol. 90, 1968, pp.109-114.
[5]Herwig, H. and Schäfer, P., Influence of variable properties on the stability of two-dimensional boundary layers, J. Fluid Mech., Vol. 243, 1992, pp.1-14.
[6]Schäfer, P., Severin, J. and Herwig, H., The effect of heat transfer on the stability of laminar boundary layers, Int. J. Heat Mass Transfer, Vol. 38 No. 10, 1995, pp.1855-1863.
[7] Special Course on Stability and Transition of Laminar Flow, AGARD-R-709.
[8]Schlichting, H., Boundary Layer Theory, McGraw-Hill, 1979.
[9]Cebeci, T. and Cousteix, J., “Modelling and Computation of Boundary-Layer Flows”, Horizons Publishing, 1999.