Solitary-wave Solution of Turbulence with Application to Benard Convection

Ishibashi, K.*, Tsuge, S,**, Nakagawa, T.M.S***         | Other Recent Papers |

Abstract:   A modified classical statistical theory of turbulence designed to include inhomogeneous turbulence is applied to high Rayleigh number regime (Ra > 5x105) of Benard convection with infinite aspect ratio. The fluctuation correlations are solved using the methods of variable separation in the form of solitary waves in a transformed space. Measurable quantities, e.g., turbulent intensities are obtained through a simple integration of the solitary-wave solution. The mean temperature distribution is calculated, and thereby the Nusselt number is determined as a function of the Rayleigh number. The power spectrum is also obtained through an inverse-Fourier transform of the solitary-wave solution. These quantities are compared with data from existing experiments. The agreement is satisfactory considering that the theory does not include any empirical parameters, as contrast to k-e models or eddy viscosity concepts.


View Figures by Clicking Each Title

Fig.1:   Solitary wave solutions for separated fluctuation variable fa in the (z, s)-plane near the lower limit of the turbulent region; Ra = 5.0 X 105 Pr=6.1, (f4; temperature fluctuation)

Fig.2:   Turbulence intensity profiles of horizontal and vertical components of the velocity. Included are the temperature profiles at three different Rayleigh numbers and for the Prandtl number corresponding to the water. The experimental data used in the comparison are available only for half the medium despite the apparent asymmetry of the solution.

Fig.3:   The mean temperature distribution at Ra = 3.97 x 106 and Pr=6.1 (water) calculated from Eq.(18) plotted against vertical distance z, and compared with experiment for water (ref.8).

Fig.4:   Power spectrum of the temperature fluctuation calculated using formula by expression (25), plotted against the nondimensional frequency, and compared with experiment( ref.10 ).

Fig.5:   The Nusselt number against the Rayleigh number   (Nu;the nondimensional heat transfer) calculated using (17) and plotted against the Rayleigh number (Ra;the nondimensional temperature difference between walls) on the original figure from ref.5.



1. INTRODUCTION

The statistical theory of turbulence emerged from the classical work of Karman and Howarth[1]. This was originally limited to homogeneous, isotropic turbulence, but it has since been revised to include inhomogeneous turbulence [2,3,4]. Two key issues to make this approach practical are:

i) Introduction of a method of separation of variables which converts equations for multi-point correlations into those for respective points [5].
ii) Devising a closure condition which allows for the separation with full nonlinear terms included.

These key findings have led to a set of equations that govern the mean flow and mean fluctuations respectively, coupled each other through turbulent transports.

This paper describes a method for predicting the turbulence characteristics of Benard convection with an infinite aspect ratio for Rayleigh numbers higher than 5 x 105. The Benard problem is an example with enough experimental data available for comparison, and is also known as an example that is a colorful kaleidoscope [5] of flow regimes which changes with the controlling parameter. These regimes are, in order of increasing Rayleigh number, thermally strained quiescence, buoyancy driven convection, oscillation, chaos, transition to stochastic regime, and turbulence.

The area of interest in this paper is the last regime: The objective here is to solve the equations derived in ref.3 with a few simplifying assumptions based on physical insights which also are supported by flow visualizations (Sec.2.). The system of equations is solved by CFD which provides the solution of the solitary wave (Sec.3.). These solutions are integrated to give turbulent fluctuations and transfers. The quality of the solution is also discussed in the light of comparison with existing experiments (Sec.4.).


2. FORMULATION OF THE PROBLEM

The equations governing compressible turbulent flows are given in ref.3. Here we will adapt these equations to the Benard problem which can be characterized by the following physical/geometrical features:


Proceed to Next Page

Click the above button to proceed

or

For Copy of the full paper, please fill the Form below

After filling out the form, press "Submit Paper Copy Request" and we will fax, e-mail or mail you a set of the copy.

Enter First Name
Last Name
Organization Name

Street Address


E-mail address
City


State/Country     Zip Code 		   
Office Phone                             Fax Number


Remarks

                                     


            Other abstracts available on the Internet - Click to see any of the followings:

  Solitary-wave Solution of Turbulence with Application to Benard Convection
  First principle basis of the direct numerical simulation for turbulence of inert and reactive gases
  A solitary-wave representation of turbulence in the physical-plus-eddy space
  Coherent solitary-wave of mixing layer turbulence in the physical-plus-eddy space
  A New Hierarchy System on the Basis of a Master Boltzmann Equation for Microscopic Density
  Molecular and turbulent transports competing in premixed flames

Or directly obtain the sources by visiting the following URLs:

FOR Title : First principle basis of the direct numerical simulation for turbulence of inert and reactive gases
Location : http://xxx.lanl.gov/abs/chao-dyn/9712020
FOR Title : A solitary-wave representation of turbulence in the physical-plus-eddy space
Location : http://xxx.lanl.gov/abs/chao-dyn/9803013
FOR Title : Coherent solitary-wave of mixing layer turbulence in the physical-plus-eddy space
Location : http://xxx.lanl.gov/abs/chao-dyn/9803035


* Tecnhical Research and Development Institute, JDA, 1-2-24, Ikejiri, Setagaya-ku, Tokyo 154, Japan

** TARA Center, University of Tsukuba, Tsukuba 305, Japan

*** Kanazawa Institute of Technology, Ishikawa 921, Japan

To be published in Proceedings of Eighth International Colloquium on Differential Equations (Editor; D.Bainov, International Science Publishers, 1997)  keywords including Solitary turbulent wave, turbulence theory, Benard turbulence, turbulence with no eddy viscosity model, temperature inversion.


For further information: STachibana@lbl.gov




















This is a portion of the full scientific paper copy of "Solitary-wave Solution of Turbulence with Application to Benard Convection by Tsuge, Ishibasi and Nakagawa". The contents, keywords and conventional approaches include Turbulence, modelings, computational, theoretical fluid dynamics, Turbulence modeling, Temperature inversion, High Rayleigh number Benard convection, Turbulent temperature fluctuations, fluid dynamics, turbulent fluctuations, not eddy viscosity modelling, Solitary wave solution, solitary-wave, compressible turbulent flows, turbulent theory, analytical, scientific, engineering, theoretical and academic research without any empirical parameters like eddy viscosity concepts and approaches.