# Turbulent Flow and Heat Transfer

## 1. Introduction to Heat Transfer

• ### Conduction

• At 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 medium
• For conduction, it requires a medium (fluid and solid)
• Temperature gradient across the medium is the main driving force for transfer of heat energy. Hence, the rate of heat transfer is calculated using 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 fans or pumps. It is called forced convection. If bulk motion is naturally by density difference of fluid (buoyancy) which is natural convection

• Radiation is 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  ### 1.2 Detials 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.  ### 1) Forced Flow :

• The flow is driven by external means like fan, pump and blower etc.
• Due to high velocity flow, the heat removal rate is by forced convection compared to natural convection ### 2) Free flow /Natural flow

• Flow is driven by the density difference between hot and cold fluids ### 1.3) Richardson Number

• 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.   ## 2. Features of Turbulent Flows

• Fluid flows are classifies 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

### 2.1 Characteristics of Laminar flow (smooth)

• Low velocity
• Less fluctuations in flows (Smooth flow)
• Viscosity dominated flows ### 2.2 Characteristics of Turbulent Flow

Turbulent flow physics is characterized with 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
• Large scale of coherent eddy structures in turbulence depends on boundary conditions and geometry, whereas small eddy structure is more universal  • ### Instantaneous Quantity

• All instantaneous quantities vary with times and space
• These quantities are averaged with time or both space and time as per 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 ### 2.3 Critical Reynolds number

• Relative magnitude of inertial and viscous terms is Reynolds number • Increasing 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<Recrit
• Turbulent flow: Re>Recrit  ## 3. Time Average and Instantaneous Velocities

• If we recorded the velocity at a particular point in the real (turbulent) fluid flow, the instantaneous velocity (U) would look like this:  • 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

### 3.1 Averaged Continuity equation: • X-Momentum equation ### 3.2 Reynolds Averaged Navier Stokes (RANS) equations • Reynolds Stress Tensor • To close the problem, we need additional equations to solve
• Reynolds stresses : total 9 – 6 are unknown
• Total 4 equations and 4 + 6 = 10 unknowns
• 10 unknown= Three velocities + pressure + six Reynolds-stresses

## 4. Modeling of Turbulent Flow using 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 Rij ### 4.1 RANS Modeling using The Closure Problem

• The Reynolds Stress tensor must be solved • The RANS equations can be closed in two ways:
• Approach 1 : Without EV ### 4.2 Modeling Based of Turbulent (eddy) Viscosity

Boussinesq Hypothesis:

Reynolds shear stress (anisotropic part of Rij) is proportional to mean velocity strain rate tensor ### 4.3 Laminar vs Turbulent Viscosity

• μ is Fluid property and often called laminar viscosity
• μt 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 vt = μt
• The turbulent viscosity is isotropic or anisotropic

### 4.4 Modeling of Turbulent Viscosity and Its Scales

• The models can be used to predict the turbulent viscosity:

1) Turbulent Scales Related to k and ε

• Characteristics of the Turbulent Structures   • Turbulent intensity is measured in terms of percentage (%)

### 4.5 The 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

### 4.6 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 ### 4.7 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 models for boundary layer flows. Hence, this model is useful for flow over blunt bodies or in separated flows
• Wilcox’ 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 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 freestream
• SST k-ω model is a better option than standard k-ε and k-ω models ## 5. Modelling 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 DT = turbulent thermal diffusivity

### 5.1 Computation of Heat Transfer Coefficients Heat transfer coefficient

• The heat transfer coefficient is computed with a fixed bulk temperature.
• Note that reference temperature must be specified first. • Turbulent Boundary Layer • 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 for, the heat transfer coefficient is higher at the leading of a flat surface or any inlet. This is because of higher temperature difference between fluid and wall at 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 ### 6. Case Studies of Forced Flow

6.1 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 used solve governing equation 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 %

### 6.2 Cooling of Office 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 ## 7. 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.

### 7.1 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 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.

### 7.2 Modeling of Turbulent Flow in Natural Convection

• For turbulent modelling, 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

7.3 Turbulence Generation Due to Buoyancy

• The generation of turbulent kinetic energy due to buoyancy (Gb) is added in the TKE equation.
• For turbulent kinetic energy equation, Gb is  an 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. ## 8. Case Study for Natural Convection

### 8.1 Door and roof vents on a building with heated wall:

• 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: ### 8.2 Natural flow circulation predicted by the CFD simulation ## 9. 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 modelling, averaged mass, mass and momentum equations are solved to get averaged quantities
• Example of mixed convection: Passenger Cabin in an Aircraft ### 3 thoughts on “Turbulent Heat Transfer and Its CFD Modeling”

1. Great Blog. Information, Diagrams, Equations all are mindblowing. Wonderful.

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