For any fluid flow conditions flowing on a plate, inside tubes, channel flows or conduits, whether natural or forced, laminar flow or turbulent flow conditions are found and the specific equations are used to evaluate heat transfer. However, even though if transition flow is arrived, discussed or given, from literature, for transition flow conditions, the equations are not seen. My question is for conditions of transition flow, are there any equations to solve for heat transfer?
In a 1-d sense, i.e. if the wall values like the wall heat flux depend only on the streamwise direction x, the heat coefficient c_h (Stanton number) is proportional to the friction coefficient c_f (Reynolds analogy). For the both exist analytically derived formulas for laminar flow, and semi-empirical formulas for turbulent flow. During transition, the c_f goes up relatively quickly, often overshooting for a small streamwise stretch the turbulent value (by up to about 20-50%), and finally settles to the turbulent value. As for the heat flux, there is an additional driver by the higher adiabatic wall temperature (for Mach number greater than, say, 0.5 relevant). Strictly taken, the Reynolds analogy does not hold for transitional flow, but is still a good indicator if not exploited very locally.
The overshoot is often a consequence of (quasi-)steady disturbances like longitudinal vortices or velocity streaks that have large amplitude and generate an additional mean-flow distortion (average in spanwsie direction z) that increases the wall shear and heat flux in excess of the mean-flow alteration by the 2-d Reynolds stress. Those disturbances may be also present in a steady laminar flow, then also increasing c_f somewhat.
In a 2-d sense, if the heat flux is measured or computed from the actual flow field and is a function of x and z, there may be regions where the heat flux is lower or stronger than in the spanwise average. This is significant if those (quasi-steady) disturbances imprint a steady structure on the wall shear distribution: the c_f is larger where fast fluid is brought to the wall, and lower where wall fluid is brought up, similarly for c_h. These regions can only be obtained by the timewise average of the full 3-d flow field.
The behavior of transition flow is complex but you can found many equations for this phenomena
In a 1-d sense, i.e. if the wall values like the wall heat flux depend only on the streamwise direction x, the heat coefficient c_h (Stanton number) is proportional to the friction coefficient c_f (Reynolds analogy). For the both exist analytically derived formulas for laminar flow, and semi-empirical formulas for turbulent flow. During transition, the c_f goes up relatively quickly, often overshooting for a small streamwise stretch the turbulent value (by up to about 20-50%), and finally settles to the turbulent value. As for the heat flux, there is an additional driver by the higher adiabatic wall temperature (for Mach number greater than, say, 0.5 relevant). Strictly taken, the Reynolds analogy does not hold for transitional flow, but is still a good indicator if not exploited very locally.
The overshoot is often a consequence of (quasi-)steady disturbances like longitudinal vortices or velocity streaks that have large amplitude and generate an additional mean-flow distortion (average in spanwsie direction z) that increases the wall shear and heat flux in excess of the mean-flow alteration by the 2-d Reynolds stress. Those disturbances may be also present in a steady laminar flow, then also increasing c_f somewhat.
In a 2-d sense, if the heat flux is measured or computed from the actual flow field and is a function of x and z, there may be regions where the heat flux is lower or stronger than in the spanwise average. This is significant if those (quasi-steady) disturbances imprint a steady structure on the wall shear distribution: the c_f is larger where fast fluid is brought to the wall, and lower where wall fluid is brought up, similarly for c_h. These regions can only be obtained by the timewise average of the full 3-d flow field.
Dear Pisipaty
The freely available textbook by Lienhard and Lienhard has been a valuable resource for me on this topic.
http://web.mit.edu/lienhard/www/ahtt.html
All the best
Johan
Thanks Abbas J Jubear , Markus J. Kloker and Johan Zietsman for providing useful answers.
As far as I know, there are no equations available for transition flows. Transition is complex in nature. Equations are available only for laminar and turbulent flows.
Transition flow is tricky in nature. The flow is not stabilized and difficult to analyze. However appropriately conditions near to laminar or turbulent flow, equations are used to get a near ideal solution.
Transition flow exists but getting or deriving equations is complex in nature.
Transition regions vary with the flow conditions with different cross sections. This changes continuously and does not remain steady. The equations are only framed for steady conditions only and development of equations in transition zone, I think is some what difficult. However attempt can be made on local conditions.
As far as my knowledge, there are no equations developed for transition flow. The equations are available for laminar and turbulent flows.
Transition zone is quite not stable. Equations are only valid for a steady flow. As I understand, research is on the way for transition zone and till now generalized equations are not developed in transition zone.
Transition flow is quite different and peculiar to estimate. This condition is not steady and the equations to be framed based on localised approaches.
Laminar flows and turbulent flows equations are existing and one is acquainted to solve the problems. Transition flow is critical and one has to consider several conditions before arriving to transition condition. Transition flow is typical and if one has to go into, a person has to consider local conditions existing at that point.
Transition flow is critical and difficult to estimate. Rare is known to tackle in this area. In order to study, this should be based on local conditions. The equations are yet to be developed in detail.
Transition flow is typical in both fluid mechanics, fluid dynamics and heat transfer problems. The problem has to be necessarily done and hence are taken up near laminar or turbulent flow conditions and complete formulae for transition flow is yet to be known clearly.
Transition is described in literature as how it comes as it will be in between laminar and turbulent flows. Effort was on to develop equations. However, this transition flow varies and depends on several conditions or characteristics. Hence this is unclear.
Flows can be characterized by laminar, transition and turbulent. Transition flow comes in between laminar and turbulent and each and every flow should pass from laminar through transition to turbulent This flow is peculiar and difficult to understand because of its fluctuations. Hence transition flow is considered at local conditions and the way of developing equations are going on.
Transition flow behaviour is difficult to estimate and the process should be defined. This is different for the equipment considered. Hence equations to derive may definitely vary from equipment to equipment and depend on various parameters on study.
Transition flow comes in between laminar and turbulent flows. Generally transition is considered in forced convection and depends on the cross section in which the fluids are following. Due to the complexity of flow, transition flow equations are on the way.
As many of the researchers are giving equations and considering effects of laminar and turbulent flow behaviour, an attempt now is currently focussed on transition flow which is critical or typical. This condition varies with equipment to equipment.
Transition flow is unstable and exists between the conversion from laminar to turbulent. Experiments are done to predict transition. The equations for this are on developing stage.
Transition flow is difficult to understand and this varies depending on the flow through or on certain cross sections. This could be analyzed at local conditions and the transition equations are in developing stage.
Transition flow occurs in between laminar and turbulent flows. Not much described of its nature in literature, but research is going on to find the equations under transition conditions.
Laminar flows and turbulent flows are in general dealt in many heat transfer problems. In between laminar and turbulent flows, transition flow occurs and can be done for local conditions under some cases. Not all equations for transition flow are derived but are in developing stage.
Transition means in between. Regular flows are laminar and turbulent and equations for determining these are existing. The typical case that is transition, can be evaluated for a certain case and for certain condition. However, generally, the transition flow equations are in developing stage.
In evaluating heat transfer, two conditions in general are taken, one is laminar and other turbulent. They are easy to do and analyze. All the others are also taken near to these regions. Strictly, the evaluation of transition flow equations are in developing stage and some of them can be evaluated or done under local conditions.
Transition flow comes in between laminar and turbulent flows and the equations are successfully derived for laminar and turbulent flow conditions. Transition flow conditions are estimated and can be tried and solved under local conditions and the equations are in the developing stage.
Transition flow is typical. When the flow transforms from laminar to turbulent, flow should cross transition. The equations are in developing stage and the people are trying out at local conditions.
Transition flows comes in behaviour of flow of fluids in fluid mechanics and as well to evaluate their heat flow using equations in heat transfer. Compared to laminar flow or turbulent flow, transition flow is difficult to evaluate since this neither comes to laminar or turbulent and equations are defined for these flows only. Research is going on to find the equations for transition knowing their flow behaviour.
As far as my knowledge, transition flow, till now, analysis are made and almost no equations are developed. The reason being, the flow might be fluctuating. Hence, transition conditions can be tried for local conditions initially and applied to other conditions later.
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