Criteria Of Turbulent Transition In Parallel Flows

📝 Abstract

Based on the energy gradient method, criteria for turbulent transition are proposed for pressure driven flow and shear driven flow, respectively. For pressure driven flow, the necessary and sufficient condition for turbulent transition is the presence of the velocity inflection point in the averaged flow. For shear driven flow, the necessary and sufficient condition for turbulent transition is the existence of zero velocity gradient in the averaged flow profile. It is shown that turbulent transition can be effected via a singularity of the energy gradient function which may be associated with the chaotic attractor in dynamic system. The role of disturbance in the transition is also clarified in causing the energy gradient function to approach the singularity. Finally, it is interesting that turbulence can be controlled by modulating the distribution of the energy gradient function.

💡 Analysis

Based on the energy gradient method, criteria for turbulent transition are proposed for pressure driven flow and shear driven flow, respectively. For pressure driven flow, the necessary and sufficient condition for turbulent transition is the presence of the velocity inflection point in the averaged flow. For shear driven flow, the necessary and sufficient condition for turbulent transition is the existence of zero velocity gradient in the averaged flow profile. It is shown that turbulent transition can be effected via a singularity of the energy gradient function which may be associated with the chaotic attractor in dynamic system. The role of disturbance in the transition is also clarified in causing the energy gradient function to approach the singularity. Finally, it is interesting that turbulence can be controlled by modulating the distribution of the energy gradient function.

📄 Content

1 The Third International Symposium on Physics of Fluids (ISPF3) 15-18 June, 2009, Jiuzhaigou, China CRITERIA OF TURBULENT TRANSITION IN PARALLEL FLOWS

HUA-SHU DOU Temasek Laboratories, National University of Singapore, Singapore 117508 tsldh@nus.edu.sg; huashudou@yahoo.com BOO CHEONG KHOO Department of Mechanical Engineering, National University of Singapore, Singapore 119260 mpekbc@nus.edu.sg
Received 1 May 2009

Abstract Based on the energy gradient method, criteria for turbulent transition are proposed for pressure driven flow and shear driven flow, respectively. For pressure driven flow, the necessary and sufficient condition for turbulent transition is the presence of the velocity inflection point in the averaged flow. For shear driven flow, the necessary and sufficient condition for turbulent transition is the existence of zero velocity gradient in the averaged flow profile. It is shown that turbulent transition can be effected via a singularity of the energy gradient function which may be associated with the chaotic attractor in dynamic system. The role of disturbance in the transition is also clarified in causing the energy gradient function to approach the singularity. Finally, it is interesting that turbulence can be controlled by modulating the distribution of the energy gradient function.

Keywords: Turbulent Transition, Criteria, Energy gradient, Role of disturbance, Singularity.

2

1. Introduction Turbulence is one of the most difficult problems in classical physics and mechanics. The origin of turbulence is an intensive research topic for many years. It has been observed in experiments that turbulent transition depends on the Reynolds number and the amplitude of disturbance. The disturbance amplitude is found to scale with Re by an exponent of -1 with transversal injection disturbance [1]. This phenomenon has been successfully modeled by the energy gradient method [2-5]. There is good agreement between the energy gradient method and experiments for parallel flows in three aspects: the scaling of the disturbance threshold with Re, the location of the maximum disturbance in the transition and the critical value of the energy gradient function. It also explained the possible mechanism of receptivity to free-stream disturbance and the mechanism of self- sustenance of turbulence in boundary layer flows which is absent in pipe and channel flows. In this study, based on the energy gradient method [2-5], criteria for turbulent transition in parallel flow are proposed for both pressure driven and shear driven flows.

2. Energy Gradient Theory: Re-visited Dou [2] proposed an energy gradient method for the purpose of clarifying the mechanism of flow instability and turbulent transition. It is demonstrated that the relative magnitude of the total mechanical energy of fluid particles gained and the energy loss along streamline due to viscous friction in a disturbance cycle determines the disturbance amplification or decay. For a given base flow, a stability criterion is expressed as [2-5],
   Const U v K m  ' ,
   ) / ( ) / ( s H n E K      ,

(1) where K is a dimensionless field variable (function) and expresses the ratio of the gradient of the total mechanical energy in the transverse direction and the loss of the total mechanical energy in the streamwise direction. Here, 2 5.0 V p E    is the total mechanical energy per unit volumetric fluid, s is along the streamwise direction, n is along the transverse direction, H is the energy loss per unit volumetric fluid along the streamline, ρ is the fluid density, u is the streamwise velocity of main flow, U is the

3 characteristic velocity, and m v' is the amplitude of disturbance velocity. Since the magnitude of K is proportional to the global Reynolds number (  UL  Re ) for a given geometry [2], the criterion of Eq.(1) can be written as [3-5],
Const U v m  ' Re or
1 (Re) ~ ) ' (  c m U v .

(2) This scaling has been confirmed by careful experiments observed for pipe flow and boundary layer flow, and this result is in agreement with the asymptotic analysis of the Navier-Stokes equations (for Re) [3-5].

In Eq.(1), K corresponds to the base flow where the flow parameters are calculated for the smooth laminar flow at given Re. When a disturbance with finite amplitude is imposed on the flow, the distribution of K of the averaged flow is different from that of the smooth laminar flow without disturbance imposed.

3. Criteria for Turbulent Transition As is well known, a laminar flow is smooth in which the motion of fluid can be described by deterministic equations whereas a turbulent flow is of disorder and chaotic. The question is how a smooth laminar flow becomes turbulent/chaotic flow in which the motion of fluid becomes largely non-deterministic? What is the role of disturbance in the transition? From Eq.(1), it is observed

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