What is ReTheta

In this post I’m going to talk about ReTheta. ReTheta (Reθ) appears in the Transition SST turbulence model in FLUENT, but the FLUENT theory guide doesn’t go into detail about ReTheta. This blog entry isn’t intended to give a great level of detail either, but to serve as a bit of background for those who are in my position, seeing this thing pop-up and thinking “what’s that ?”.

Reθ is the Momentum thickness Reynolds number.  

Re is a Reynolds number based on the momentum thickness as the size dimension. Momentum thickness a way of measuring boundary layer thickness*.

equation 1: momentum thickness Reynolds number.

equation 2: momentum thickness. where: ρ = density, u = velocity in the direction of flow, subscript infinity indicates bulk flow conditions.

Reθ in the Transition SST Model

I do not at this current moment in time have a full understanding of the transition SST model. But from what I’ve read so-far, my understanding is:

• The Transition SST model includes several Reynolds numbers, but the most important is the “Transition momentum thickness Reynolds number” Re~θt.

• The Transition SST model calculates Reθt on bulk properties, making Reθt a function of freestream conditions. Reθt is a function of turbulent intensity Tu and Thwaites’ pressure gradient coefficient λ. In the Fluent Theory Guide this function is withheld as propriety, other sources (Menter 2006, Nichols 2010) show the same type of empirical equations for Reθt. “Local Correlation Based Transition Modelling” (LCTM) aims to get round the issue of previous empirical transition models, which pose a numerical/programming issue for general purpose CFD due to nonlocal formulations.

• This is in the model as a transport equation, given in the Fluent Theory Guide available online. The transport equation models diffusion of Re~θt into the boundary layer. Pθt the source term in this equation, is designed to match Re~θt to the value of Reθt found empirically.

• Re~θt in the boundary layer then causes transition through the various mechanisms. Reθt is the critical Reynolds number where intermittency first starts to increase in the boundary layer. Reθc is a similar empirical function of Re~θt. Reθc is proportional Reθt is the location where the velocity first starts to deviate from the laminar profile.

That’s my, very basic, understanding of ReTheta. I will update this post as my understanding of it improves

 

 

 

*the momentum thickness is “the distance by which the surface would have to be moved parallel to itself towards the reference plane in an ideal fluid stream of velocity u0 to give the same volumetric flow as occurs between the surface and the reference plane in the real fluid” – David Balmer

Ref:

Balmer, D. (2014). Integral Momentum Method for Solution of Boundary Layer Problems. online: http://www.see.ed.ac.uk/~johnc/teaching/fluidmechanics4/2003-04/fluids10/integral.html [accessed 17 Feb 2015]

Menter, FS. Et al. (2006). A Correlation-Based Transition Model Using Local Variables – Part 1: Model Formulation. Journal of Turbomachinery. 128. p. 413-422

Nichols, RH. (2010). Turbulence Models and Their Application to Complex Flows. chapter 11:Boundary Layer Transition Simulation. University of Alabama at Birmingham.

Transport Phenomena

The key-concept behind the study of transport phenomena is the idea that heat transfer, mass transfer and fluid flow are analogous. In some cases, they can be described in a way where they resemble each other in their mathematics/algebra. This is what I want to cover today. In other cases a 'heat-and-mass transfer analogy' such as the 'J-factor' or 'Chilton-Colburn analogy' is used to calculate, say, mass transfer based on empirical heat tranfer correllations, and I may cover this type of approach in future blogs.

Undergraduate chemical engineering courses often introduce the concept of mass transfer by using a heat transfer analogy. A comparison might be drawn between Fick's law for mass transfer and Fourier's law for heat transfer. The student learns that "the driving force for heat transfer is temperature difference, the driving force for mass transfer is concentration difference*". As the student learns more about mass transfer and heat transfer the similarities between the two can often become more-and-more opaque. 

But with CFD, we're often interested in problems where mass transfer, heat transfer and fluid flow are influencing each other. Many chemical engineering problems fall into this category. Instead a generalised representation of transport equations can be made. Any variable, whether it's concentration, velocity or enthalpy/temperature, can be represented by phi (Φ).

Equation 1: Generalised transport equation. Where the variables rho (ρ), t, uj, xj have their usual meanings (density, time, xj = distance, uj = velocity, j defines direction (1,2,3) ). Capital gamma (Γ) = diffusivity coefficient, SΦ = The source term(s) for variable Φ.

1. The term (ρΦ) denotes the amount of extensive property available in a unit volume. This rate-of-change is only important for transient-phenomenon. ∂(ρΦ)/∂t is called the transient term, and accounts for the accumulation of Φ inside the control volume (chemical engineers will recognise the connotations between accumulation and unsteady-state mass balance!).
2. ∂(ρujΦ)/∂xj is called the convection term.  uj is the velocity, it can be thought of as a flow per unit area. If ∂xj is the length of the control volume, then the convection term is like the amount of Φ being brought into & out-of the control volume by the flow.
3. The term Γ represents diffusion. For mass transfer in laminar flow, this is just molecular diffusion. For heat, this is heat conduction in the fluid. For fluid flow, this is viscosity (“viscosity is the diffusion of momentum”). This is where turbulence can make a big difference. In laminar flows Γ can be measured by experiments, say, in a lab environment. In turbulent flows these become highly dependent on the flow and this is no longer possible – this is where turbulence modelling comes in.
4. Source terms account for any creation or destruction of the variable Φ. For mass transfer, this may be used to describe reactions consuming reactants & generating products. But it is not limited to chemical transformations – source terms would also describe phase change or adsorption. For heat transfer similarly, this may represent heat transfer on a wall or from a phase change for example. In momentum transfer, pressure gradient ∂P/∂xj is a source term. Any term that can’t be cast as a convection or diffusion term is considered a source term.

	Φ	Γ	SΦ net source
Conservation of mass	1	0	0
Mass transfer	ω	ρDeff	R
Momentum	ui	μeff	-∂P/∂xj + ρBi +Sui
Energy (enthalpy)	h	keff/Cp	Q

Table 1: where omega ω = mass fraction, Deff effective diffusivity, R = rate of generation of species (eg: by reaction), ui velocity in direction i(1,2,3), µeff = effective viscosity, -∂P/∂xj = pressure gradient, ρBi = buoyancy forces, centrifugal forces, particle drag etc. (Sui may represent eg: Stoke’s Stress Laws), h = enthalpy, keff = thermal conductivity of fluid, Cp = heat capacity of fluid, Q is a heat source, such as heat of a reaction.

The Transport equations are a fundamental part of CFD. Whilst the concept that heat, mass and momentum are similar is much older, the representation in Equation 1, the generalised transport equation, is attributed to the work of Prof. DB Spalding and Prof. SV Patankar, who are also responsible for the SIMPLE algorithm. The equations are often written out in full, but at the moment I’m finding the table-version a little easier-on-the-eyes.

* I prefer to think of this phrase instead as "the driving force for mass transfer is the concentration's 'distance-to-equilibrium' ". I find this helps when thinking about problems such as two-film theory, where the concentration difference at equilibrium for say, a separation oil-and-water, is defined by the water-octanol partition coefficient. 

References

Date, AW. 2005. Introduction to Computational Fluid Dynamics. Cambridge: Cambridge University Press.

Kim, D. Kim, D. Lee, KS. 2015. Frosting Model for Predicting Macroscopic and Local Frost Behaviours on a Cold Plate. International Journal of Heat and Mass Transfer. 82. p.135-142.

First Post - CFD Yourself Manifesto

In this first post, I'm going to cover the basic purpose of my blog.

I'm a UK student currently studying for a doctorate in engineering. Computational fluid dynamics (CFD) is a huge part of my current research. But my background (my undergraduate and master's) is in chemical engineering. With the exeption of a four-day introductory course, I have no formal education in CFD, or with numerically* solving partial differentail equations (PDE) generally, which is not something chemical engineers generally do at the u-grad level in much detail. So I'm looking to start this blog to engage the subject and show a bit of initiative in learning it, to practice writing about CFD, and hopefully to help other people out by providing useful titbits for the beginner in CFD. A lot of what I want to cover will be obvious to many with a taught background & experience in CFD, (maybe any readers that fall into this catagory can help me out if you spot any mistakes or have any suggestions :) ? ). CFD Yourself will act as a general notebook on my journey teaching myself in CFD, I don't plan to follow any predefined structure or schedule, and I'm just going to blog about what pops into my head.

CFD Yourself will be about:

• Engaging the subject of CFD
• Generating titbits of basic info on CFD
• Getting into the online community

Alright, so I hope you enjoy my blogging !


* a quick-flick through my old u-grad coursenotes reveals I did infact cover the algebra of PDEs, but that was six years ago, (doesn't time fly !).