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 high-velocity 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, three-dimensional, 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 non-linear (inertia) term in the Navier Stokes (NS) equations. This non-linearity 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 in-compressible 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 time-averaged from both sides

• X-Momentum 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 Reynolds-stresses

 

 Modeling of Turbulent Flow with RANS

• After decomposing the velocity into mean and instantaneous parts and time-averaging, the instantaneous Navier-Stokes equations may be rewritten as the Reynolds-averaged Navier-Stokes (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 k-ε model

• The most widely-used engineering turbulence model for industrial applications
• The k-ε 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 k-ε 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, m²/s²,

ε= dissipation rate= turbulent kinetic energy/time, m2/s³

• To find the turbulent viscosity we need two equations which are given below

Standard k-ε 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 k-ω Model

• Shear Stress Transport (SST) Model
• The SST model is a hybrid two-equation model that combines the advantages of both k-ε and k-ω models
• The k-ω model performs much better than k-ε model for boundary layer flows. Hence, this model is useful for flow over blunt bodies or in separated flows
• Wilcox’s original k-ω model is overly sensitive to the free stream value (BC) of ω, while the k-ε model is not prone to such a problem
• The k-ε and k-ω models are blended such that the SST model functions like the k-ω close to the wall and the k-e model in the free stream
• SST k-ω model is a better option than the standard k-ε and k-ω models

Modelings of Turbulent Heat Transfer

• Instantaneous equation of mass momentum and energy equations
• After time-averaging 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 k-ω turbulence model
  • Pressure discretization utilized the standard method with SIMPLEC coupling
  • QUICK discretization scheme for momentum-ω 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 ( Nuconv )				% 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 Z-direction 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 temperature-dependent 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