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.
In this problem, the Navier-Stokes equations reduce to a second order, linear, ordinary differential equation (ODE),
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
3.1.2. 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.
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.