Table of Contents
CFD Modeling of Turbulent Heat Transfer: Forced, Natural and Mixed Convection
by Dr. Sharad Pachpute
Introduction to Heat Transfer
Mode of Heat Transfer
Conduction

 At the molecular level, transfer of energy by transfer of electron and molecular vibration. At a classical level it is interpreted as thermal diffusion without bulk motion of the medium
 For conduction, it requires a medium (fluid and solid)
 The temperature gradient across the medium is the main driving force for the transfer of heat energy. Hence, the rate of heat transfer is calculated using the Fourier Law of Heat Conduction
Convection

 Heat is transferred by bulk fluid motion
 Newton’s law of cooling is used to find the convective heat transfer
 In convection, the bulk fluid motion which is carried out by external means of fans or pumps. It is called forced convection. If the bulk motion is naturally by the density difference of fluid (buoyancy) which is natural convection
Radiation

 Radiation is the transfer of energy in terms of electromagnetic waves (emission). Hence this mode does not medium for its transport
 Emissive properties of surface and surface temperature are used to find the radiative heat transfer using the Stefan Law
Details of Convective Heat Transfer

 Heat transfer through a fluid is by convection in the
 Presence of bulk fluid motion and by conduction in the absence of it
Forced Convection
 The flow is driven by external means like a fan, pump, blower, etc.
 Due to highvelocity flow, the heat removal rate is by forced convection compared to natural convection
Free flow (Natural flow)
 Flow is driven by the density difference between hot and cold fluids
Role of Richardson Number in Convective Heat Transfer
 To determine whether the flow is driven by natural convection, forced convection, or both, we examine the Richardson number.
 Richardson number (Ri) represents the relative magnitude of natural convection effects to forced convection effects.
Features of Turbulent Flows
 Fluid flows are classified into two major categories: a) Laminar flow, b) turbulent flow
 Understanding the characteristics of turbulent flow is essential before going to turbulent convective heat transfer
Characteristics of Laminar flow (smooth)
 Low velocity
 Less fluctuations in flows (Smooth flow)
 Viscosity dominated flows
Characteristics of Turbulent Flow
Turbulent flow physics is characterized by unsteady, threedimensional, irregular, stochastic motion in which transported quantities (mass, momentum, scalar species) fluctuate in time and space
 Enhanced mixing of these quantities results from the fluctuations
 Unpredictability
 Eddies/rotating fluid masses
 Turbulent flows contain a wide range of eddy sizes
 The large scale of coherent eddy structures in turbulence depends on boundary conditions and geometry, whereas small eddy structure is more universal
Instantaneous Quantity

 In turbulent flow, all instantaneous quantities vary with times and space
 These quantities are averaged with time or both space and time as per the modeling of flows
 Reynolds Averaged Navier Stokes (RANS) equations are derived by substituting the averaged velocity and pressure in instantaneous governing equations in turbulent flows
Critical Reynolds number
 The relative magnitude of inertial and viscous terms is Reynolds number
 Increasing the Reynolds number increases the nonlinear (inertia) term in the Navier Stokes (NS) equations. This nonlinearity term is sensitive in the NS solution to fluctuations
 Laminar flow: Re<Re_{crit}
 Turbulent flow: Re>Re_{crit }
Reynolds Decomposition for Averaged Equations
 If we recorded the velocity at a particular point in the real (turbulent) fluid flow, the instantaneous velocity (U) would look like this:
 The average quantity is obtained by integrating the instantaneous quantity over the period of time T
 At any point in time, instantaneous velocity
 The time average of any fluctuating quantity is zero:
 Note that the root mean square (RMS) of fluctuating is not zero
 The turbulent energy (k) is calculated using the fluctuating velocity components as:
 Reynolds Decomposition is the decomposition of instantaneous variables
 Substitute into Steady incompressible Navier Stokes equations to get averaged governing equations
 The instantaneous conservation equations for mass and momentum are substituted with the above Reynolds decomposition. After it, the overall equation is timeaveraged from both sides
 XMomentum equation
Reynolds Averaged Navier Stokes (RANS) equations
 Reynolds Stress Tensor
 To close the problem, we need additional equations to solve
 Reynolds stresses: 6 stresses from a total of 9 stresses are unknown
 Total 4 other equations are needed to find mean quantities U, V, W, P
 Total 10 unknowns variables = Three velocities + Pressure + Six Reynoldsstresses
Modeling of Turbulent Flow with RANS
 After decomposing the velocity into mean and instantaneous parts and timeaveraging, the instantaneous NavierStokes equations may be rewritten as the Reynoldsaveraged NavierStokes (RANS) equations:
 The Reynolds stresses are additional unknowns introduced by the averaging procedure, hence they must be modeled (related to the averaged flow quantities) in order to close the system of governing equations
 Total 6 unknowns in the RSS tensor R_{ij}
RANS Modeling using The Closure Problem

 The Reynolds stress tensor must be solved
 The RANS equations can be closed in two ways:
Modeling of Turbulent With Eddy Viscosity
 In this approach, equations for all six Reynolds stresses are solved along with an additional equation of turbulent dissipation rate (ε)
Modeling of Turbulent Flow With Eddy Viscosity Hypothesis
Boussinesq Hypothesis
Reynolds shear stress (anisotropic part of Rij) is proportional to mean velocity strain rate tensor
Laminar vs Turbulent Viscosity
 μ is Fluid property and often called laminar viscosity
 μ_{t} is Flow property and termed as turbulent viscosity
 The unit of turbulent viscosity is Pas which is same as that of molecular (laminar) viscosity
 The kinematic turbulent (eddy) viscosity is defined as v_{t} = μ_{t }/ρ
 This turbulent viscosity can be isotropic or anisotropic
Modeling of Turbulent Viscosity and Its Scales
 The models can be used to predict the turbulent viscosity:
Turbulent Scales Related to k and ε

 Characteristics of the Turbulent Structures
 Turbulent intensity is measured in terms of percentage (%)
Two Equation kε model
 The most widelyused engineering turbulence model for industrial applications
 The kε model is based on the turbulent (fluctuating) kinetic energy (k) per unit mass and its dissipate rate (ε)
 The instantaneous kinetic energy (K) of turbulent flow is the sum of mean flow kinetic energy (K), and turbulent (fluctuating flow) kinetic energy k)
 In the kε model, the eddy viscosity is calculated the fluctuating velocity and length scale which are depends on the turbulent kinetic energy (k) per unit mass and its dissipate rate (ε)
k =turbulent kinetic energy, m^{2}/s^{2},
ε= dissipation rate= turbulent kinetic energy/time, m2/s^{3}
 To find the turbulent viscosity we need two equations which are given below
Standard kε Model
 Apart from Reynolds averaged mass and momentum, additional two equations are solved to find turbulent viscosity
 k Transport Equations
 ε Transport Equations
 Model Constants
 Turbulent /eddy viscosity
Two Equation SST kω Model
 Shear Stress Transport (SST) Model
 The SST model is a hybrid twoequation model that combines the advantages of both kε and kω models
 The kω model performs much better than kε model for boundary layer flows. Hence, this model is useful for flow over blunt bodies or in separated flows
 Wilcox’s original kω model is overly sensitive to the free stream value (BC) of ω, while the kε model is not prone to such a problem
 The kε and kω models are blended such that the SST model functions like the kω close to the wall and the ke model in the free stream
 SST kω model is a better option than the standard kε and kω models
Modelings of Turbulent Heat Transfer
 Instantaneous equation of mass momentum and energy equations
 After timeaveraging the governing equations, we get Reynolds Averaged Navier Stokes Equations
Where D_{T} = turbulent thermal diffusivity
Computation of Heat Transfer Coefficients
 The heat transfer coefficient is computed with a fixed bulk temperature.
 Note that reference temperature must be specified first.
 Turbulent Boundary Layer
 The turbulent boundary layer is fluctuating. However, the averaged boundary layer is considered to find average heat transfers as averaged mass, momentum, and energy equations are used to find the averaged velocity, pressure, and temperatures
 For laminar, the heat transfer coefficient is higher at the leading of a flat surface or any inlet. This is because of the higher temperature difference between fluid and wall at the inlet
 As the temperature difference decreases, heat transfer decreases with increasing the distance.
 A transition in flows increases the heat transfer. Then after it decreases due an increase in temperature difference between the fluid and wall
Case Studies for Modeling of Forced Flow
Cooling of Heated Cylinder
 The following problem of cold air flowing over a heated cylinder was carried out
 CFD results are presented and compared with the experimental data
 Temperature contoured: Red color (high value), Blue color (low value)
ANSYS FLUENT is used to solve governing equations sequentially.
 Laminar flow :
 Pressure discretization utilized the standard method with PISO coupling
 The second order Upwind discretization for Momentum ,energy equation
 Turbulent flow :
 SST kω turbulence model
 Pressure discretization utilized the standard method with SIMPLEC coupling
 QUICK discretization scheme for momentumω equation and energy equations
 CFD Results
 The average heat transfer obtained from numerical simulations are compared with available experimental data
Re  Tu
% 
Convective Nusselt number ( Nu_{conv} )  % difference
From Scholton et al. 

Present numerical value  K. Szczepanik et. al  Scholton et.al. (experimental)  Zukauskas  
7190  1.6  55.22  67.3 (steady k model )  51  47.3  8.88% 
50350  0.36  171.28  191.1 (steady k model )  155.1  151.7  10.56 % 
Cooling of Room by a Ceiling Fan
 Using CFD modeling we can get find the region of high velocity for better comfort for seating of people in the office
 Using simulation we can find the number of fans required for thermal comfort in the hall
Natural Convection
 The fluid flow without external means is called a natural flow
 Whenever there is a large difference in temperature of fluid and surface, the difference in density is the main driving force for fluid flows
 The direction of flow can be up, down or inclined depending on the orientation of geometries and direction of buoyancy
 Laminar vs Turbulent Flow
 Momentum equation along the Zdirection of gravity
 In some CFD solver (ANSYS FLUENT), the variable change is calculated for the pressure fields when the gravity is considered
 The momentum equation results with an additional source or sink of momentum due to buoyancy
Where P* is the static pressure used by CFD solver to avoid round of errors for boundary conditions.
The Boussinesq Approximation for Natural Convection
 Boussinesq model assumes the fluid density is constant in all terms of the momentum equation except the body force term
 In the body force term, the fluid density is linear.
 For many natural convection problems, this treatment provides faster convergence than other temperaturedependent density descriptions.
 The assumption of constant density reduces the nonlinear nature of the governing equations.
 The Boussinesq assumption is valid when density variations are small. Cannot be used with species transport or reacting flows.
Modeling of Turbulent Flow in Natural Convection
 For turbulent modeling, averaged mass, mass, and momentum equations are solved to get averaged quantities.
 The additional terms in the momentum equations like Reynolds stress is modeled using the eddy viscosity model
 One more additional term due to buoyancy which is modeled in the following section
Turbulence Generation Due to Buoyancy
 The generation of turbulent kinetic energy due to buoyancy (Gb) is added in the TKE equation.
 For the turbulent kinetic energy equation, G_{b} is included in the k equation as a source of turbulence by buoyancy for other turbulence models. Buoyancy effects can be considered in the k–ω models
 Source term due to Buoyancy
 Generation of turbulent kinetic energy due to buoyancy (Gb) is by default neglected in the dissipation (TKE),ε equation.
Case Study for Natural Convection
Door and roof vents on a building with heated walls:
 The roof static pressure is set to 0 while the door static pressure must be given a hydrostatic head profile based on the height of the building.
 The boundary conditions for pressure are set as
 The above equations can be written as
 Note: In this case, if you can set the operating density equal to the external ambient density then the hydrostatic component can be ignored:
 Natural flow circulation predicted by the CFD simulation
Mixed Convection
 It is a combination of both Free and Forced Convection
 In mixed convection flows, the effect of free convection is not negligible
 Governing Equations for mass, momentum, and energy (instantaneous)
 For turbulent modeling, averaged mass, mass, and momentum equations are solved to get averaged quantities
 Example of mixed convection: Passenger Cabin in an Aircraft
Summary
 Convection (bulk fluid) flow can be driven by forced mode (mechanical means) or natural mode (density difference)
 The Richardson number determines the relation between forced and natural convection
 Turbulence heat flux is modeled based on the mean temperature gradient
 The turbulent viscosity and Prandtl number are used to calculate the thermal eddy diffusivity. The value of Prandtl number is taken as 0.9 in most cases
 The Boussinesq approximation is used to determine the Buoyancy term in the momentum equation
 CFD Modeling predicts well the turbulent heat transfer for both forced and natural circulation
Great Blog. Information, Diagrams, Equations all are mindblowing. Wonderful.
Thanks for valuable comments
In this paper, Computational fluid dynamics (CFD) modeling of turbulent heat transfer behavior of Magnesium Oxidewater nanofluid in a circular tube was studied. The modeling was two dimensional under k–? turbulence model. The base fluid was pure water and the volume fraction of nanoparticles in the base fluid was 0.0625%, 0.125%, 0.25%, 0.5% and 1%. The applied Reynolds number range was 3000–19000. Three individual models including single phase, Volume of Fluid (VOF) and mixture were used.
The results showed that the simulated data were in good agreement with the experimental ones available in the literature. According to the experimental work (literature) and simulation (this research), Nusselt number (Nu) increased with increasing the volume fraction of nanofluid. However friction factor of nanofluid increased but its effect was ignorable compared with the Nu on heat transfer increment.
It was concluded that two phase models were more accurate than the others for heat transfer prediction particularly in the higher volume fractions of nanoparticle. The average deviation from experimental data for single phase model was about 11% whereas it was around 2% for two phase models.
Excellent