3.1. FLD01: Steady, 2D Poiseuille flow

3.1.1. Description

Plane Poiseuille flow is defined as a steady, laminar flow of a viscous fluid between two horizontal parallel plates separated by a distance, \(H\). Flow is induced by a pressure gradient across the length of the plates, \(L\), and is characterized by a 2D parabolic velocity profile symmetric about the horizontal mid-plane as illustrated in Fig. 3.1.

../_images/fld01-setup.png

Fig. 3.1 Plane Poiseuille flow between two flat plates of length L, separated by a distance H.

In this problem, the Navier-Stokes equations reduce to a second order, linear, ordinary differential equation (ODE),

(3.1)\[\mu_{g}\frac{d^{2}u_{g}}{dy^{2}} = \frac{dP_{g}}{\text{dx}},\]

where \(\mu_{g}\) and \(P_{g}\) are correspondingly the fluid viscosity and pressure, and \(u_{g}\) and \(v_{g}\) are respectively the \(x\) and \(y\) velocity components. Furthermore, it is assumed that gravitational forces are negligible, the pressure gradient is constant, i.e., \(dP_{g}/dx = C\), and all velocity components are zero at the channel walls. The resulting analytical solution to Eq.3.1 is given as

(3.2)\[u_{g}\left( y \right) = - \frac{dP_{g}}{\text{dx}}\frac{1}{2\mu_{g}}y\left( H - y \right).\]

3.1.2. Setup

########################################################################
#                                                                      #
# Author: Aniruddha Choudhary                           Date: Jan 2015 #
# Horizontal channel (rectangular plane Poiseuille flow)               #
#                                                                      #
# A pressure gradient is imposed over x-axis cyclic boundaries. The    #
# north and south walls are no-slip.                                   #
#                                                                      #
########################################################################

  RUN_NAME = 'FLD01'
  DESCRIPTION = 'Steady, 2D Poiseuille Flow'

#_______________________________________________________________________
# RUN CONTROL SECTION

  UNITS = 'SI'
  RUN_TYPE = 'NEW'


  ENERGY_EQ =     .F.
  SPECIES_EQ(0) = .F.

  GRAVITY = 0.0

  CALL_USR = .T.

#_______________________________________________________________________
# NUMERICAL SECTION

  MAX_NIT =    200000
  TOL_RESID =   1.0E-10

  LEQ_PC(1:9) = 9*'NONE'

  DISCRETIZE(1:9) = 9*2
  NORM_G = 0.0

#_______________________________________________________________________
# GEOMETRY SECTION

  COORDINATES = 'CARTESIAN'

  ZLENGTH = 1.00     NO_K = .T.
  XLENGTH = 0.20     IMAX = 8
  YLENGTH = 0.01     JMAX = 8

#_______________________________________________________________________
# GAS-PHASE SECTION

  RO_g0 = 1.0     ! (kg/m3)
  MU_g0 = 1.0d-3  ! (Pa.s)

#_______________________________________________________________________
# SOLIDS-PHASE SECTION

  MMAX = 0

#_______________________________________________________________________
# INITIAL CONDITIONS SECTION

  IC_X_w(1) =    0.00    ! (m)
  IC_X_e(1) =    0.20    ! (m)
  IC_Y_s(1) =    0.00    ! (m)
  IC_Y_n(1) =    0.01    ! (m)

  IC_EP_g(1) =   1.0

  IC_P_g(1) =    0.0     ! (Pa)

  IC_U_g(1) =   10.0     ! (m/sec)
  IC_V_g(1) =    0.0     ! (m/sec)

#_______________________________________________________________________
# BOUNDARY CONDITIONS SECTION

! Inlet and outlet: Periodic BC
!---------------------------------------------------------------------//

  CYCLIC_X_PD = .T.
  DELP_X =     240.00    ! (Pa)

! Top and bottom walls: No-slip
!---------------------------------------------------------------------//
! Bottom wall
  BC_X_w(3) =    0.00    ! (m)
  BC_X_e(3) =    0.20    ! (m)
  BC_Y_s(3) =    0.00    ! (m)
  BC_Y_n(3) =    0.00    ! (m)

  BC_TYPE(3) = 'NSW'

! Top wall
  BC_X_w(4) =    0.00    ! (m)
  BC_X_e(4) =    0.20    ! (m)
  BC_Y_s(4) =    0.01    ! (m)
  BC_Y_n(4) =    0.01    ! (m)

  BC_TYPE(4)  = 'NSW'

#_______________________________________________________________________
# OUTPUT CONTROL SECTION

  RES_DT =        1.0    ! (sec)
  SPX_DT(1:9) = 9*1.0    ! (sec)

  FULL_LOG = .T.

  RESID_STRING  = 'P0', 'U0', 'V0'

#_______________________________________________________________________
# DMP SETUP

!  NODESI =  1  NODESJ =  1  NODESK =  1

3.1.3. Results

The analytical and numerical solutions for x-velocity, \(u_{g}\), are shown in Fig. 3.2. Only a subset of the numerical solution data points are plotted causing the appearance of a slight shift in presented data points. The observed error demonstrates a second-order rate of convergence with respect to grid size in the y-axial direction. This is attributed to the second-order discretization of the viscous stress term as convection/diffusion terms do not contribute to the solution.

../_images/image331.png

Fig. 3.2 Steady, 2D channel flow x-velocity profile (left), absolute error in x-velocity solution (center), and observed order of accuracy (right) using four grid levels (JMAX = 8, 16, 32, 64).

The fluid pressure, \(P_{g}\), varies linearly along the length of the plates as shown in Fig. 3.3. The largest observed absolute error is bounded above by \(10^{- 12}\) and occurs for the finest mesh. This error is attributed to the convergence criteria of the linear equation system.

../_images/image341.png

Fig. 3.3 Steady, 2D channel flow pressure profile (left) and absolute error in pressure solution (right) using four grid levels (IMAX = 8, 16, 32, 64).