®
COMSOL Multiphysics

Laminar Flow ModelingLaminar FlowSingle-Phase Flow Equations
 Navier–Stokes, continuity, and energy equation,

 where,𝐷𝐷𝒖𝒖= −𝛁𝛁𝛁𝛁 + 𝛁𝛁 � 2𝜇𝜇𝑺𝑺 + 𝑭𝑭𝜌𝜌
𝐷𝐷𝐷𝐷𝐷𝐷𝜌𝜌+ 𝜌𝜌𝛁𝛁 � 𝒖𝒖 = 0𝐷𝐷𝑡𝑡
𝐷𝐷𝐷𝐷
𝐷𝐷𝐷𝐷
𝜌𝜌𝐶𝐶𝑝𝑝= 𝛻𝛻 � 𝑘𝑘𝑘𝑘𝑘𝑘 + 2𝜇𝜇𝑺𝑺: 𝑺𝑺 + 𝛼𝛼𝑝𝑝 𝑇𝑇+ 𝑄𝑄
𝐷𝐷𝐷𝐷
𝐷𝐷𝐷𝐷𝐷𝐷𝜕𝜕= + 𝒖𝒖 � 𝛻𝛻,
𝐷𝐷𝐷𝐷 𝜕𝜕𝜕𝜕

12T
𝑺𝑺 =𝛁𝛁𝛁𝛁 + 𝛁𝛁𝛁𝛁 − 𝛁𝛁 � 𝒖𝒖 𝐈𝐈23

Special Case: Perfect Gas
 For a perfect gas,
𝑝𝑝 = 𝜌𝜌𝜌𝜌𝜌𝜌 Using the energy equation, the continuity equation may be recast as,
1 𝐷𝐷𝐷𝐷 𝛻𝛻 � 𝑘𝑘𝑘𝑘𝑘𝑘
2𝜇𝜇𝑺𝑺: 𝑺𝑺
𝑄𝑄++++ 𝛻𝛻 � 𝒖𝒖 = 0𝛾𝛾𝑝𝑝 𝐷𝐷𝐷𝐷𝜌𝜌𝐶𝐶𝑝𝑝 𝑇𝑇𝜌𝜌𝐶𝐶𝑝𝑝 𝑇𝑇 𝜌𝜌𝐶𝐶𝑝𝑝 𝑇𝑇

 Introducing scales for the dependent variables, 𝑈𝑈0 , 𝑇𝑇0 , 𝑝𝑝0 , properties, 𝜌𝜌0 , 𝜇𝜇0 , 𝑘𝑘0 , and 𝐿𝐿0 (length), and𝐿𝐿0 /𝑈𝑈0 (time). Scale pressure changes with dynamic pressure, ∆𝑝𝑝~𝜌𝜌0 𝑈𝑈02 ,1 𝐷𝐷𝐷𝐷𝛾𝛾𝛾𝛾 𝐷𝐷𝐷𝐷

𝑈𝑈0~𝑀𝑀𝑀𝑀2𝐿𝐿0𝛻𝛻 � 𝑘𝑘𝑘𝑘𝑘𝑘+
𝜌𝜌𝐶𝐶𝑝𝑝 𝑇𝑇
1~
𝑃𝑃𝑃𝑃

2𝜇𝜇𝑺𝑺: 𝑺𝑺+𝜌𝜌𝐶𝐶𝑝𝑝 𝑇𝑇(𝛾𝛾 − 1)𝑀𝑀𝑀𝑀2~𝑅𝑅𝑅𝑅

𝑄𝑄++ 𝛻𝛻 � 𝒖𝒖 = 0𝜌𝜌𝐶𝐶𝑝𝑝 𝑇𝑇1 𝑄𝑄𝐿𝐿20~~1 = 0
𝑃𝑃𝑃𝑃 𝑘𝑘0 𝑇𝑇0Dimensionless Numbers
 Similarly, for the momentum equation,

𝜌𝜌0 𝑈𝑈02
𝐿𝐿0𝐷𝐷𝒖𝒖+ 𝛁𝛁𝛁𝛁 − 𝛁𝛁 � 2𝜇𝜇𝑺𝑺 − 𝑭𝑭 = 0𝐷𝐷𝐷𝐷
1~1 = 0~1~1~
𝑅𝑅𝑅𝑅

𝜌𝜌

 The dimensionless numbers are,Reynolds number (inertial forces/viscous forces):

𝑅𝑅𝑅𝑅 =

Mach number (flow speed/speed of sound in fluid): 𝑀𝑀𝑀𝑀 =Peclet number (convection/conduction):

𝑃𝑃𝑃𝑃 =

𝜌𝜌0 𝑈𝑈0 𝐿𝐿0𝜇𝜇0𝑈𝑈0,
𝑎𝑎0𝑎𝑎0 =𝜌𝜌0 𝐶𝐶𝑝𝑝 𝑈𝑈0 𝐿𝐿0𝑘𝑘0𝛾𝛾𝑝𝑝0𝜌𝜌0Determining the Flow Regime, Re At low 𝑅𝑅𝑅𝑅 (≪ 1): creeping (Stokes) flow

Viscous forces tend to damp out all flow disturbances
Reversible smooth flow pattern At intermediate 𝑅𝑅𝑅𝑅 ~1 − 2000∗ : laminar flow
Inertial forces become increasingly important

Viscous forces are confined to boundary layers, shear layers, and wakes
Regular, smooth flow pattern At high 𝑅𝑅𝑅𝑅 (> 4000∗ ): turbulent flow

Flow disturbances grow by nonlinear interactions producing a cascade of eddies (vortices)
Viscous damping is active everywhere but only has a significant effect on thesmallest eddies
Disordered (chaotic) flow pattern*Critical Re of 2000 has been mentioned above as an example considering an internal channel flow

Reynolds Number Examples𝑥𝑥

𝑈𝑈0 𝑥𝑥𝑅𝑅𝑅𝑅𝑥𝑥 =𝜈𝜈
𝑈𝑈0 𝐿𝐿0𝑅𝑅𝑅𝑅𝐿𝐿 =𝜈𝜈

𝑈𝑈0 𝐷𝐷𝐻𝐻
4𝐴𝐴, 𝐷𝐷𝐻𝐻 ≡𝑅𝑅𝑅𝑅𝐷𝐷ℎ =𝜈𝜈𝑃𝑃
𝑈𝑈0 𝐷𝐷𝑅𝑅𝑅𝑅𝐷𝐷 =𝜈𝜈

𝐿𝐿0𝐷𝐷

Determining the Flow Regime, Ma
 For 𝑀𝑀𝑀𝑀 = 0: formally incompressible flow

Speed of sound is infinite (hypothetical case), instantaneous spread of pressure disturbances For 0 < 𝑀𝑀𝑀𝑀 < 0.3: weakly compressible flow
Density changes due to pressure by max. 5%Density changes can also occur due to the dependency on temperature For 𝑀𝑀𝑀𝑀 > 0.3: compressible flow

Thermodynamic effects not negligible
Transonic, supersonic flow with shock waves

Choosing the Right Fluid Flow Interface
𝑴𝑴𝑴𝑴 < 𝟎𝟎. 𝟑𝟑
𝑅𝑅𝑅𝑅 ≪ 1
𝑅𝑅𝑅𝑅~1 − 𝑅𝑅𝑒𝑒𝑐𝑐𝑐𝑐
𝑅𝑅𝑅𝑅 > 𝑅𝑅𝑒𝑒𝑐𝑐𝑐𝑐
𝑅𝑅𝑅𝑅 > 𝑅𝑅𝑒𝑒𝑐𝑐𝑐𝑐
𝑅𝑅𝑅𝑅 > 𝑅𝑅𝑒𝑒𝑐𝑐𝑐𝑐

𝑴𝑴𝑴𝑴 > 𝟎𝟎. 𝟑𝟑
𝑅𝑅𝑅𝑅 < 𝑅𝑅𝑒𝑒𝑐𝑐𝑐𝑐
𝑅𝑅𝑅𝑅 > 𝑅𝑅𝑒𝑒𝑐𝑐𝑐𝑐
𝑅𝑅𝑒𝑒 → ∞Single-Phase Flow Creeping flow also called Stokes flow
 Laminar flowNewtonian and non-Newtonian flow Turbulent flow
2 algebraic models
7 transport-equation models

 Large eddy simulation (LES)
3 variational multi-scale models

 Detached eddy simulation (DES) Rotating machinery
Laminar and turbulent flow Viscoelastic flow

The Single-Phase Flow user interfaces as displayed in
the Physics list in the CFD Module.

Single-Phase Flow: General Functionality
 Swirl flow
Includes the out-of-plane velocity component
for axisymmetric flows

 Enable porous media domains
Model porous media flow or coupled free andporous media flow

 Gravity and reduced-pressure option
 Special purpose boundary conditions
Fully developed laminar and turbulent inflow
and outflow
Wall conditions on internal shells for
simulating thin immersed structures
Screen conditions for simulating thin
perforated plates and wire gauzes

Fully developed turbulent flow condition at a star-shaped inlet.

Laminar Flow Solves the Navier–Stokes andcontinuity equations
Stationary or time dependent
Incompressible, weaklycompressible, or compressible
flow (𝑀𝑀𝑀𝑀 < 0.3)

 Neglect inertial term for Re <<1 Enable porous media domains
 Include gravity
 Define reference pressure level
and reference temperature

Laminar Flow node and Settings window

DemoOutlet

Model Definition Laminar flow in water Fully developed flow at the inlet
 Pressure condition at the outlet
 No-slip conditions at walls
 Symmetry conditions at the twolateral surfaces
 Why?
Typical expansion found in manysystems, e.g. in medtech
Benchmark with flow separation

Inlet

Walls

Symmetry

SymmetryDue to symmetry, we only have to model one eight of the model domain, provided that the flow is steady andthat the inlet flow is perpendicular to the inlet boundary.

About Fluid Pressure Flow is driven by pressure gradients
 Absolute pressure needed for material
property evaluation; e.g., 𝜌𝜌(𝑝𝑝𝐴𝐴 , 𝑇𝑇)
 Generally two ways to calculate thepressure:

Case 1: ∆𝑝𝑝 ~ 𝑝𝑝𝐴𝐴  set 𝑝𝑝𝑟𝑟𝑟𝑟𝑟𝑟 = 0 and solve forthe absolute pressure directly 𝑝𝑝𝐴𝐴 = 𝑝𝑝

Case 2: ∆𝑝𝑝 ≪ 𝑝𝑝𝐴𝐴  set 𝑝𝑝𝑟𝑟𝑟𝑟𝑟𝑟 close to typicalsystem pressure level and solve for the (small)gauge pressure only 𝑝𝑝𝐴𝐴 = 𝑝𝑝𝑟𝑟𝑟𝑟𝑟𝑟 + 𝑝𝑝

 Why is it important?  solver stability &convergenceSection of the settings window inthe fluid flow interface

Gravity Property
 Includes a volume force on alldomains,
𝐅𝐅 = 𝜌𝜌𝒈𝒈

 Option to use reduced pressure forincompressible flow Boussinesq approximation available
for nonisothermal flow

Lock-release gravity current

Fluid Properties Density
From material
User defined Constitutive relation
Newtonian𝜏𝜏 = 𝜇𝜇𝛾𝛾̇ = 𝜇𝜇𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕• Dynamic viscosity

Inelastic non-Newtonian• Inelastic model

Newtonian and non-Newtonian constitutiverelations

Initial Values
 Stationary
Initial values = initial guess fornumerical solver
No physical meaning but
starting point for iterations
Boundary conditions do nothave to be consistent with initialvalues

Initial Values node and Settings window

Initial Values
 Time dependent
Initial values = initial state of theflow and pressure fields
All boundary conditions must be
consistent with initial condition(at t = 0)
• Use a converged stationary solution
as initial condition for the timedependent study• Alternatively, use the trivial initialcondition (u, p = 0) and ramp up theflow with time at the boundaries
Trivial initial values require that the flow is ramped up
with time at the boundaries, here with a step function.

Wall Wall conditions:
No slip
Slip
Electroosmotic velocity
Slip velocity
Leaking wall
Navier slip
 Wall movement can be picked up
automatically when a moving frame isdefined in the model

Wall node and Settings windowLocation of Outlet
Boundary Extend the domain:
Avoid recirculation and othercomplex flow patterns at the outlet

Flow in a semicircular pipewith a 90-degree bendFully DevelopedFlow

𝐿𝐿𝐸𝐸 = 0.05𝐷𝐷𝐻𝐻

𝑈𝑈𝐷𝐷𝐻𝐻𝜈𝜈 Available for inlets and outlets
 Equations for fully developed flow
are solved at the inlet or outlet
boundary

Flow in a semicircular pipe with and without the Fully
Developed Flow option

Interior Wall
 Avoid meshing of thin structuresby using interior boundaries
 Slip, no slip, moving wall
 Allows discontinuities (velocity,pressure) across the boundary

Modeled as surfaces in 3D and edges in 2D

Screen “Screen” refers to a barrier withperforations, e.g. wire gauze,grille, or perforated plate Assumed to have a width, which issmall compared to the resolved
length scales:Can be modeled as an interior edge
(2D) or interior surface (3D)

 Common correlations forresistance and refraction
coefficients are included

Pressure (color table) and streamlines for pipe flow
obstructed by a screen inclined at 45 degrees

Fan, Interior Fan,Grille, Vacuum Pump
 Represented using lumped curves
as boundary conditions
 Boundary can only be either aninlet or an outlet
 Fan manufacturers provide curves
for the static pressure as afunction of flow rate

Define the flow direction on exterior and interior
boundaries, and fan curve (or similar)

Symmetry

Symmetry,
Axisymmetry, andPeriodicity

Symmetry
Symmetry

Axisymmetry

Periodicity

Periodicity

 Reduce memory requirement andcomputational time Physics, materials, and geometry
must all be symmetric

Although the physics, materials, and geometry are symmetric, there may be asymmetric solutions. Here is aclassical example of the Karman vortex street. There is a symmetric but very unstable solution.