FULLY-DEVeloped flows.


  1. Introduction
  2. Single-slab fully-developed flows
  3. Periodic fully-developed flows
  4. Exemplification
  5. Sources of further information

1. Introduction

PHOENICS provides for the simulation of fully-developed flow and heat or mass transfer inside ducts of arbitrary constant cross-section, and also for the more general case of periodically fully-developed flow in which the duct cross-sectional area varies periodically in the streamwise direction.

The case of developed flow in a duct or passage of constant cross-section will be referred to as single-slab fully-developed flow, since such flows can be simulated by performing calculations on a single-slab of computational cells.

This is useful when it is desired to compute a fully-developed pipe-, plane-walled-channel- or Couette-flow situations, without, as was formerly necessary, simulating the flow in a long duct, either elliptically or parabolically.

Periodically developed flows occur in a variety of heat-exchange devices, such as for example electronic cooling devices and corrugated heat-exchanger passages. The means by which PHOENICS can be employed to simulate these types of flows is explained below in sections 2 and 3.

2. Single-slab fully-developed flows

2.1 Description

In this type of fully-developed flow, the velocity and dimensionless temperature ( or mass concentration ) profiles are invariant with distance along the duct in regions away from the entrance.

The analysis of such problems is now possible in PHOENICS by: performing the calculation on a single IZ slab of computational cells; deactivating the axial convection terms for all solved-for variables; and deactivating the built-in pressure-gradient term in the W1 momentum equation.

Two-phase-flow simulations with ONEPHS=F are not supported.

If the duct wall is circular or an infinite plane, the flow is one- dimensional and then the pressure P1 and cross-stream velocity (U1 or V1) need not be solved. Consequently, all convection terms can be de-activated. For any other duct cross-section, the flow in the slab will be two-dimensional and so the pressure P1 must be solved and only the axial-convection terms can be deactivated.

Since the user must render the W1 pressure-gradient term inactive, it is then usually necessary to prescribe a mean axial pressure gradient in order to create finite axial velocities, so that the mass flow rate is an outcome of the PHOENICS calculation.

When it is the mass flow rate that is known and the user requires the pressure gradient, this procedure requires a number of separate calculations to be made where the pressure gradient is precribed and then adjusted from calculation to calculation until the computed mass flow rate matches the specified value.

Therefore, a more economical alternative has been provided whereby PHOENICS automatically calculates the pressure gradient from overall continuity for a given mass flow rate. The pressure adjustment procedure is based on that proposed by Patankar and Spalding [1972] for parabolic flows, and the PHOENICS implementation is describedby Madhav [1992].

Under certain heating and cooling conditions, such as constant heat flux or uniform temperature at the duct walls, a fully developed temperature profile can exist in the sense that the dimensionless temperature profile remains invariant with distance along the duct.

There are a large number of possible thermal boundary conditions for a rectangular duct, since any of the four walls may be adiabatic, of constant heat flux or constant temperature, or thermal conduction in the surrounding wall may be significant. The only cases presently supported by PHOENICS are surface heat flux constant and surface temperature constant in the peripheral direction.

In PHOENICS, the solution of the energy equation for fully-developed flow may proceed by way of the H1 or TEM1 equation. The energy equation is solved for thermally developed flow by prescribing the axial gradient of bulk enthalpy cp*dTb/dz, which for a constant wall heat flux can be obtained from an overall energy balance.

For a uniform wall temperature, cp*dTb/dz is also prescribed, which is equivalent to specifying the energy flow out of the slab. Thus, for hydrodynamically and thermally developed flow, the following volumetric source term appears in the energy equation:

wherein theta is the dimensionless temperature defined as

where Tw is the duct wall temperature and Tb is the fluid bulk temperature. For a constant wall heat flux, equation (2.1) reduces to:

whereas for a uniform wall temperature it reduces to:

The mathematical formulation of the mass transfer problem is analogous to the heat-transfer problem, in that the mass concentration replaces the temperature, and the mass diffusivity replaces the heat diffusivity.

2.2 Activation

(i) Hydrodynamics

The user may use the TERMS command to deactivate the axial pressure- gradient, and in one-dimensional flows the command may also be used to deactivate convection, e.g.

For two-dimensional problems the axial convection terms can be deactivated by the use of a Group 12 PATCH, e.g.

If the axial pressure gradient is to be prescribed, the following arbitarily-named PATCH is introduced in the Q1 file:

where GDPDZ is the user-defined axial pressure gradient. If, instead, the user wishes to prescribe the mass flow rate, then the following PIL commands need to be set in the Q1 file:

PHOENICS will then compute the axial pressure-gradient automatically, and print the result in the RESULT file. For the case when the axial pressure gradient dp/dz is specified, the following formula is useful:

where: win is the bulk velocity; rho is the fluid density; D is the hydraulic diameter; and f is the friction factor, given by:

for laminar flow;

for turbulent flow in smooth tubes; and by

for turbulent flow in fully-rough tubes. Here, Re is the Reynolds number based on hydraulic diameter and bulk velocity, and eps is the relative roughness defined as the sand-grain roughness height divided by the hydraulic diameter.

For non-Couette flows, the activation of the single-slab solver requires several TERMS, PATCH and COVAL settings; therefore PHOENICS provides a short-hand PIL command entitled FDSOLV which arranges all the necessary settings.

The first argument of this command differs according to whether the mass flow rate or axial pressure drop is to be specified by the user. The second argument is used to specify the known quantity, as follows:


(ii) Heat and Mass Transfer

The option to simulate thermally-developed flow is activated by setting FDFSOL=T in the Q1 file, and introducing the following PATCH and COVAL statement:

wherein the PATCH name = FDFCHF for a constant heat-flux boundary condition, and FDFCWT for a constant wall-temperature boundary condition.

For the latter, the wall temperature must be defined by setting the VALue equal to the wall temperature rather than GRND1. For a constant heat-flux boundary condition, the user may wish to fix the temperature at the centre-line of the duct so as to define a datum. If the temperature is not fixed in this way, the temperature variable can be interpreted as (T-Tw). For both types of boundary condition, the bulk temperature is printed in the RESULT file.

Finally, it should be noted that the option to simulate thermally- developed flow may be used irrespective of whether the automatic pressure adjustment is activated, i.e. the user may still prescribe the axial pressure-gradient in the Q1 file.

3. Periodic fully-developed flows

Periodic fully-developed flows can be simulated in PHOENICS by using the built-in x-cyclic (XCYCLE=T) option and the procedure described by Patankar et al [ 1977], provided that: the calculation does not employ cylindrical polar coordinates ( for then there is no z-cyclic option ); the lateral or transverse flow is not cyclic as well; or the flow does not have complex thermal boundary conditions.

In any of these events, the user is advised to adopt the approach of repeatedly transferring the downstream exit values of all variables, except pressure, to the inlet plane at IZ=1; as demonstrated with PHOENICS by Beale [1990] and Chang and Mills [1991]. Whichever approach is adopted, the solution domain is restricted to a single geometrical module which repeats itself in identical fashion in the actual geometry.

Following Patankar et al [1977], the pressure in fully-developed periodic flow can be decomposed into two components, as follows:-

where p' is the periodic pressure, and dp/L is the axial pressure gradient along the periodic element of the duct in the axial direction, x. Thus, in order to simulate fully-developed periodic flow in PHOENICS, the mean pressure gradient can be extracted from the by substituting equation (3.1) into the momentum-equations, which leads to the following volumetric source term in the U1 momentum equation:-

where -dp'/dx is the periodic pressure gradient (determined from the PHOENICS solution of P1 for p') and dp/L is the prescribed axial pressure gradient, which may be introduced via the following PIL commands:

This PATCH introduces the required distributed momentum source in the U1 momentum equation, so that each U1 momentum cell experiences an axial force equal to the product of the cell volume and the prescribed mean pressure gradient.

With this procedure the actual flow rate is determined by the pressure gradient, so that if the mass flow rate is known the user must adjust the pressure gradient from calculation to calculation so that the converged solution corresponds the desired value of the mass flow rate.

The iterative adjustment of the pressure gradient for a new flow rate Q, might be coded in GROUND using the following formulae, which is appropriate for turbulent flow:

where alf is a relaxation factor, and the quantities in square brackets denote last-iterate values.

The initial value of dp/L may be estimated from equation (2.5) above by considering a force balance between the pressure-gradient term and the wall-friction forces in a constant cross-section duct operating at the desired mass flow rate.

For a certain class of thermal boundary conditions, the temperature field can also become periodically fully-developed in some special way. Such a facility is not a presently available as a switch-on option in PHOENICS, although the methodology employed by Patankar et al [1977] and Patankar and Prakash [1981] can be introduced by the user through GROUND, as demonstrated for example by Prakash [1985].

4. Exemplification

The following library cases illustrate the use of the single-slab fully-developed-flow feature;

quasi-one-dimensional (ie fully-developed 2D)

quasi-two-dimensional (ie fully-developed 3D)

An example can also be found in the one-phase section of the "active-demo" series, and many other examples involving different turbulence models can be found in the advanced turbulence models section of the input library.

Library case 257 demonstrates the method of using periodic boundary conditions for a fully-developed 2D channel duct flow.

5. Sources of further information

The FORTRAN coding associated with the fully-developed flow facility may be found in subroutine GXFDFS.

'Basic heat transfer', Chapters 6,7 & 9, McGraw Hill, Int. Student Edition, (1977).
Patankar, S.V. and Spalding, D.B.,
'A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows', Int.J.Heat Mass Transfer, 15, p1787, (1972).
Madhav, M.T.,
'A numerical algorithm for the solution of fully- developed flows', MSc Thesis, University of Greenwich, (1992).
S.V.Patankar, C.H.Liu and E.M.Sparrow,
'Fully-developed flow and heat transfer in ducts having streamwise-periodic variations of cross-sectional area', ASME J. Heat Transfer, Vol.99, p180, (1977).
S.V.Patankar and C.Prakash,
'An analysis of the effect of plate thickness on laminar flow and heat-transfer in interrupted-plate passages', Int.J.Heat Mass Transfer, Vol.24, No.11, p1801, (1981).
' Computation of laminar flow and heat transfer in a corrugated duct passage', PDR/CHAM NA/6, CHAM, London, (1985).
' Laminar fully-developed flow and heat transfer in an offset rectangular plate-fin surface', PHOENICS Journal, Vol.3, No.1, p1, (1990).
B.H.Chang and A.F.Mills,
' Application of a low-Reynolds-number turbulence model to flow in a tube with repeated rectangular rib roughness', PHOENICS Journal, Vol.4, No.3, p262, (1991).