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Approximate Solutions of the Navier-Stokes Equation

FLUID DYNAMICSMaster Degree Programme in Physics - UNITSPhysics of the Earth and of the Environment

FABIO ROMANELLIDepartment of Mathematics & Geosciences

University of [email protected]

https://moodle2.units.it/course/view.php?id=5449

mailto:[email protected]

mailto:[email protected]

http://moodle2.units.it/course/view.php?id=2729

http://moodle2.units.it/course/view.php?id=2729

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Objectives

Appreciate why approximations are necessary, and know when and where to use.Understand effects of lack of inertial terms in the creeping flow approximation.Understand superposition as a method for solving potential flow.Predict boundary layer thickness and other boundary layer properties.

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Introduction

We derived the NSE and developed several exact solutions.In this section, we will study several methods for simplifying the NSE, which permit use of mathematical analysis and solution

These approximations often hold for certain regions of the flow field.

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Nondimensionalization of the NSE

Purpose: Order-of-magnitude analysis of the terms in the NSE, which is necessary for simplification and approximate solutions.We begin with the incompressible NSE

Each term is dimensional, and each variable or property (ρ, V, t, μ, etc.) is also dimensional.What are the primary dimensions of each term in the NSE equation?

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To nondimensionalize, we choose scaling parameters as follows

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Next, we define nondimensional variables, using the scaling parameters in Table 10-1

To plug the nondimensional variables into the NSE, we need to first rearrange the equations in terms of the dimensional variables

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Now we substitute into the NSE to obtain

Every additive term has primary dimensions {m1L-2t-2}. To nondimensionalize, we multiply every term by L/(ρV2), which has primary dimensions {m-1L2t2}, so that the dimensions cancel. After rearrangement,

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Each of terms in [ ] is a nondimensional group of parameters (Pi Group):

Strouhal number Euler numberInverse of Froudenumber squared

Inverse of Reynoldsnumber

Navier-Stokes equation in nondimensional form

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Nondimensionalization vs. NormalizationNSE are now nondimensional, but not necessarily normalized. What is the difference?

Nondimensionalization concerns only the dimensions of the equation - we can use any value of scaling parameters L, V, etc.

Normalization is more restrictive than nondimensionalization. To normalize the equation, we must choose scaling parameters L,V, etc. that are appropriate for the flow being analyzed, such that all nondimensional variables are of order of magnitude unity, i.e., their minimum and maximum values are close to 1.0.

If we have properly normalized the NSE, we can compare the relative importance of the terms in the equation by comparing the relative magnitudes of the nondimensional parameters

St, Eu, Fr, and Re.

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Comments about CD-NSE

The nondimensionalized continuity equation contains no additional dimensionless parameters. The order of magnitude of the nondimensional variables is unity if they are nondimensionalized using a length, speed, frequency, etc., that are characteristic of the flow field.

The relative importance of the terms in depends only on the relative magnitudes of the dimensionless parameters

Dynamic similarity between a model and a prototype requires all four of Par to be the same for the model and the prototype.If the flow is steady then the first term on the left side of then disappears. If the characteristic frequency f is very small such that St<<1, the flow is called quasi-steady. The effect of gravity is important only in flows with free-surface effects.

If no free surface the only effect of gravity on the flow dynamics is a hydrostatic pressure distribution in the vertical direction superposed on the pressure field due to the fluid flow.

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Creeping Flow

Also known as “Stokes Flow” or “Low Reynolds number flow”Occurs when Re << 1ρ, V, or L are very small, e.g., micro-organisms, MEMS, nano-tech, particles, bubbles

μ is very large, e.g., honey, lavag effect is negligibleSteady flowAdvective term is negligible

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Creeping Flow

To simplify NSE, assume St ~ 1, Fr ~ 1

SincePressureforces

Viscousforces

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Creeping Flow

This is important

Very different from inertia dominated flows where

Density has completely dropped out of NSE. To demonstrate this, convert back to dimensional form:

This is now a LINEAR EQUATION which can be solved for simple geometries.

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Creeping Flow

Solution of Stokes flow is beyond the scope.

Analytical solution for flow over a sphere gives a drag coefficient which is a linear function of velocity V and viscosity m.

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Example

Although the equation for creeping flow drag on a sphere was derived for a case with Re<<1, it turns out that the approximation is reasonable up to Re ≅ 1.

A more involved calculation, including a Reynolds number correction and a correction based on the mean free path of air molecules, yields a terminal velocity of 0.110 m/s; the error of the creeping flow approximation is less than 5 percent.

Analysis We follow the step-by-step method of repeating variables discussedin Chap. 7; the details are left as an exercise. There are four parameters inthis problem (n ! 4). There are three primary dimensions: mass, length, andtime, so we set j ! 3 and use independent variables V, L, and m as ourrepeating variables. We expect only one Pi since k ! n " j ! 4 " 3 ! 1,and that Pi must equal a constant. The result is

Thus, we have shown that for creeping flow around any three-dimensionalobject, the aerodynamic drag force is simply a constant multiplied by mVL.Discussion This result is significant, because all that is left to do is find theconstant, which is a function only of the shape of the object.

Drag on a Sphere in Creeping FlowAs shown in Example 10–1, the drag force FD on a three-dimensional objectof characteristic dimension L moving under creeping flow conditions atspeed V through a fluid with viscosity m is FD ! constant # mVL. Dimen-sional analysis cannot predict the value of the constant, since it depends onthe shape and orientation of the body in the flow field.

For the particular case of a sphere, Eq. 10–11 can be solved analytically.The details are beyond the scope of this text, but can be found in graduate-level fluid mechanics books (White, 1991; Panton, 1996). It turns out thatthe constant in the drag equation is equal to 3p if L is taken as the sphere’sdiameter D (Fig. 10–13).

Drag force on a sphere in creeping flow: (10–12)

As a side note, two-thirds of this drag is due to viscous forces and the otherone-third is due to pressure forces. This confirms that the viscous terms andthe pressure terms in Eq. 10–11 are of the same order of magnitude, asmentioned previously.

EXAMPLE 10–2 Terminal Velocity of a Particle from a Volcano

A volcano has erupted, spewing stones, steam, and ash several thousand feetinto the atmosphere (Fig. 10–14). After some time, the particles begin tosettle to the ground. Consider a nearly spherical ash particle of diameter50 mm, falling in air whose temperature is "50°C and whose pressure is55 kPa. The density of the particle is 1240 kg/m3. Estimate the terminalvelocity of this particle at this altitude.

SOLUTION We are to estimate the terminal velocity of a falling ash particle.Assumptions 1 The Reynolds number is very small (we will need to verifythis assumption after we obtain the solution). 2 The particle is spherical.Properties At the given temperature and pressure, the ideal gas law gives r ! 0.8588 kg/m3. Since viscosity is a very weak function of pres-sure, we use the value at "50°C and atmospheric pressure, m ! 1.474$ 10"5 kg/m · s.Analysis We treat the problem as quasi-steady. Once the falling particlehas reached its terminal velocity, the net downward force (weight) balances

FD ! 3pmVD

FD ! constant " MVL

479CHAPTER 10

FIGURE 10–11A child trying to move in a pool of

plastic balls is analogous to amicroorganism trying to propel itself

without the benefit of inertia.

V

m

FD

D

FIGURE 10–13The aerodynamic drag on a sphere

of diameter D in creeping flow is equal to 3pmVD.

V

L

m

FD

FIGURE 10–12For creeping flow over a three-

dimensional object, the aerodynamicdrag on the object does not depend on

density, but only on speed V, somecharacteristic size of the object L,

and fluid viscosity m.

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the net upward force (aerodynamic drag ! buoyancy), as illustrated in Fig.10–15.

Downward force: (1)

The aerodynamic drag force acting on the particle is obtained from Eq.10–12, and the buoyancy force is the weight of the displaced air. Thus,

Upward force: (2)

We equate Eqs. 1 and 2, and solve for terminal velocity V,

Finally, we verify that the Reynolds number is small enough that creepingflow is an appropriate approximation,

Thus the Reynolds number is less than 1, but certainly not much less than 1.Discussion Although the equation for creeping flow drag on a sphere (Eq.10–12) was derived for a case with Re "" 1, it turns out that the approxima-tion is reasonable up to Re ≅ 1. A more involved calculation, including aReynolds number correction and a correction based on the mean free path ofair molecules, yields a terminal velocity of 0.110 m/s (Heinsohn and Cimbala,2003); the error of the creeping flow approximation is less than 5 percent.

A consequence of the disappearance of density from the equations ofmotion for creeping flow is clearly seen in Example 10–2. Namely, air den-sity is not important in any calculations except to verify that the Reynoldsnumber is small. (Note that since rair is so small compared to rparticle, thebuoyancy force could have been ignored with negligible loss of accuracy.)Suppose instead that the air density were one-half of the actual density inExample 10–2, but all other properties were unchanged. The terminal veloc-ity would be the same (to three significant digits), except that the Reynoldsnumber would be smaller by a factor of 2. Thus,

The terminal velocity of a dense, small particle in creeping flow conditions isindependent of fluid density, but highly dependent on fluid viscosity.

Since the viscosity of air varies with altitude by only about 25 percent, asmall particle settles at nearly constant speed regardless of elevation, eventhough the air density increases by more than a factor of 10 as the particlefalls from an altitude of 50,000 ft (15,000 m) to sea level.

Re #rairVDm

#(0.8588 kg$m3)(0.115 m$s)(50 % 10 & 6 m)

1.474 % 10 & 5 kg$m ' s# 0.335

# 0.115 m$s

#(50 % 10 & 6 m)2

18(1.474 % 10 & 5 kg$m ' s) [(1240 & 0.8588) kg$m3](9.81 m$s2)

V #D2

18m (rparticle & rair)g

Fup # FD ! Fbuoyancy # 3pmVD ! p D3

6 rairg

Fdown # W # p D3

6 rparticleg

480FLUID MECHANICS

Terminalvelocity

V

FIGURE 10–14Small ash particles spewed from a volcanic eruption settle slowly to the ground; the creeping flowapproximation is reasonable for this type of flow field.

V

rparticle

rair, mair

Fbuoyancy

D

FD

W

FIGURE 10–15A particle falling at a steady terminalvelocity has no acceleration; therefore,its weight is balanced by aerodynamicdrag and the buoyancy force acting onthe particle.

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Inviscid Regions of Flow

Definition: Regions where net viscous forces are negligible compared to pressure and/or inertia forces

~0 if Re large

Euler Equation

The Euler equation approximation is appropriate in high Reynolds number

regions of the flow, where net viscous forces are negligible, far away

from walls and wakes.

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Inviscid Regions of Flow

Euler equation often used in aerodynamicsElimination of viscous term changes PDE from mixed elliptic-hyperbolic to hyperbolic. This affects the type of analytical and computational tools used to solve the equations.Must “relax” wall boundary condition from no-slip to slip

No-slip BCu = v = w = 0

Slip BCτw = 0, Vn = 0

Vn = normal velocity

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Irrotational Flow Approximation

Irrotational approximation: vorticity is negligibly small

In general, inviscid regions are also irrotational, but there are situations where inviscid flow are rotational, e.g., solid body rotation

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What are the implications of irrotational approximation. Look at continuity and momentum equations.Continuity equation

Use the vector identity

Since the flow is irrotational

φ is a scalar potential function

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Therefore, regions of irrotational flow are also called regions of potential flow.

From the definition of the gradient operator ∇

Substituting into the continuity equation gives

Cartesian

Cylindrical

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This means we only need to solve 1 linear scalar equation to determine all 3 components of velocity!

Luckily, the Laplace equation appears in numerous fields of science, engineering, and mathematics. This means there are well developed tools for solving this equation.

Laplace Equation

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Momentum equationIf we can compute 𝜙 from the Laplace equation (which came from continuity) and velocity from the definition , why do we need the NSE? ⇒ To

compute Pressure.

To begin analysis, apply irrotational approximation to viscous term of the NSE:

= 0

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Therefore, the NSE reduces to the Euler equation for irrotational flow:

Instead of integrating to find P, use vector identity to derive Bernoulli equation

nondimensional

dimensional

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This allows the steady Euler equation to be written as

This form of Bernoulli equation is valid for inviscid and irrotational flow since we’ve shown that NSE reduces to the Euler equation.

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However,

Inviscid

Irrotational (ζ = 0)

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Therefore, the process for irrotational flowCalculate 𝜙 from Laplace equation (from continuity)

Calculate velocity from definitionCalculate pressure from Bernoulli equation (derived from momentum equation)

Valid for 3D or 2D

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Irrotational Flow Approximation - 2D Flows

For 2D flows, we can also use the streamfunctionRecall the definition of streamfunction for planar (x-y) flows

Since vorticity is zero,

This proves that the Laplace equation holds for the streamfunction and the velocity potential

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Constant values of ψ: streamlines

Constant values of φ: equipotential linesψ and φ are mutually orthogonal

ψ and φ are harmonic functions

ψ is defined by continuity; ∇2ψ results from irrotationality

φ is defined by irrotationality; ∇2φ results from continuity

Flow solution can be achieved by solving either ∇2φ or ∇2ψ, however, BC are easier to formulate for ψ.


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Similar derivation can be performed for cylindrical coordinates (except for ∇2ψ for axisymmetric flow)

Planar, cylindrical coordinates : flow is in (r,θ) plane

Axisymmetric, cylindrical coordinates : flow is in (r,z) plane

Planar Axisymmetric


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Method of Superposition1. Since ∇2φ=0 is linear, a linear combination of two or

more solutions is also a solution, e.g., if φ1 and φ2 are solutions, then (Aφ1), (A+φ1), (φ1+φ2), (Aφ1+Bφ2) are also solutions

2. Also true for y in 2D flows (∇2ψ =0)3. Velocity components are also additive


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Given the principal of superposition, there are several elementary planar irrotational flows which can be combined to create more complex flows.

Uniform streamLine source/sinkLine vortexDoublet


The concept of superposition is useful, but is valid only for irrotational flow fields for which the equations for f and c are linear.

One must be careful to ensure that the two flow fields you wish to add vectorially are both irrotational.

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Elementary Planar Irrotational FlowsUniform Stream

In Cartesian coordinates

Conversion to cylindrical coordinates can be achieved using the transformation

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Elementary Planar Irrotational FlowsLine Source/Sink

Potential and streamfunction are derived by observing that volume flow rate across any circle is

This gives velocity components

Building Block 2—Line Source or Line SinkOur second building block flow is a line source. Imagine a line segment oflength L parallel to the z-axis, along which fluid emerges and flows uni-formly outward in all directions normal to the line segment (Fig. 10–44).The total volume flow rate is equal to V

.. As length L approaches infinity,

the flow becomes two-dimensional in planes perpendicular to the line, andthe line from which the fluid escapes is called a line source. For an infiniteline, V

.also approaches infinity; thus, it is more convenient to consider the

volume flow rate per unit depth , V./L, called the line source strength (often

given the symbol m).A line sink is the opposite of a line source; fluid flows into the line from

all directions in planes normal to the axis of the line sink. By convention,positive V

./L signifies a line source and negative V

./L signifies a line sink.

The simplest case occurs when the line source is located at the origin ofthe xy-plane, with the line itself lying along the z-axis. In the xy-plane, theline source looks like a point at the origin from which fluid is spewed out-ward in all directions in the plane (Fig. 10–45). At any radial distance rfrom the line source, the radial velocity component ur is found by applyingconservation of mass. Namely, the entire volume flow rate per unit depthfrom the line source must pass through the circle defined by radius r. Thus,

(10–42)

Clearly, ur decreases with increasing r as we would expect. Notice also thatur is infinite at the origin since r is zero in the denominator of Eq. 10–42.We call this a singular point or a singularity—it is certainly unphysical,but keep in mind that planar irrotational flow is merely an approximation,and the line source is still useful as a building block for superposition inirrotational flow. As long as we stay away from the immediate vicinity ofthe center of the line source, the rest of the flow field produced by superpo-sition of a line source and other building block(s) may still be a good repre-sentation of a region of irrotational flow in a physically realistic flow field.

We now generate expressions for the velocity potential function and thestream function for a line source of strength V

./L. We use cylindrical coordi-

nates, beginning with Eq. 10–42 for ur and also recognizing that uu is zeroeverywhere. Using Table 10–2, the velocity components are

Line source:

To generate the stream function, we (arbitrarily) choose one of these equa-tions (we choose the second one), integrate with respect to r, and then dif-ferentiate with respect to the other variable u,

from which we integrate to obtain

Again we set the arbitrary constant of integration equal to zero, since wecan add back a constant as desired at any time without changing the flow.

f (u) !V#"L

2p u # constant

$c

$r! % uu ! 0 → c! f (u) → $c

$u! f &(u) ! rur !

V#"L

2p

ur !$f

$r!

1r $c

$u!

V#"L

2pr uu !

1r $f

$u! %

$c

$r! 0

V#

L! 2prur ur !

V#"L

2pr

496FLUID MECHANICS

Ly

x

xy-plane

z

FIGURE 10–44Fluid emerging uniformly from a finiteline segment of length L. As Lapproaches infinity, the flow becomesa line source, and the xy-plane is takenas normal to the axis of the source.

y

x

V/L r

ur

u

⋅

FIGURE 10–45Line source of strength V

./L located at

the origin in the xy-plane; the totalvolume flow rate per unit depththrough a circle of any radius r mustequal V

./L regardless of the value of r.

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Using definition of (Ur, Uθ)

These can be integrated to give φ and ψ

Equations are for a source/sinkat the origin


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If source/sink is moved to (x,y) = (a,b)


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Elementary Planar Irrotational FlowsLine Vortex

Vortex at the origin. First look at velocity components

These can be integrated to give φ and ψ

Equations are for a line vortexat the origin

𝚪 is called the Circulation or Vortex Strength

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If vortex is moved to (x,y) = (a,b)

Elementary Planar Irrotational FlowsLine Vortex

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Elementary Planar Irrotational FlowsDoublet

A doublet is a combination of a line sink and source of equal magnitude

Source

Sink

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Elementary Planar Irrotational FlowsDoublet

Adding ψ1 and ψ2 together, performing some algebra, and taking a→0 gives

K is the doublet strength

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Examples of Irrotational Flows Formed by Superposition

Superposition of sink and vortex : bathtub vortex

Sink Vortex

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Flow over a circular cylinder: Free stream + doublet

Assume body is ψ = 0 (r = a) ⇒ K = Va2

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Velocity field can be found by differentiating streamfunction

On the cylinder surface (r=a)

Normal velocity (Ur) is zero, Tangential velocity (Uθ) is non-zero ⇒slip condition.

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Compute pressure using Bernoulli equation and velocity on cylinder surface

Irrotational flow

Laminarseparation

Turbulentseparation

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Integration of surface pressure (it is symmetric in x), reveals that the DRAG is ZERO. This is known as D’Alembert’s Paradox:

For the irrotational flow approximation, the drag force on any non-lifting body of any shape immersed in a uniform stream is zeroWhy?

Viscous effects have been neglected. Viscosity and the no-slip condition are responsible for

Flow separation (which contributes to pressure drag)

Wall-shear stress (which contributes to friction drag)

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Boundary Layer (BL) Approximation

BL approximation bridges the gap between the Euler and NS equations, and between the slip and no-slip BC at the wall.Prandtl (1904) introduced the BL approximationAt a given x-location, the higher the Reynolds number, the thinner the boundary layer.

(continuity plus Navier–Stokes) throughout the whole flow field. Neverthe-less, boundary layer theory is still useful in some engineering applications,since it takes much less time to arrive at a solution. In addition, there is a lotwe can learn about the behavior of flowing fluids by studying boundary lay-ers. We stress again that boundary layer solutions are only approximationsof full Navier–Stokes solutions, and we must be careful where we apply thisor any approximation.

The key to successful application of the boundary layer approximation isthe assumption that the boundary layer is very thin. The classic example is auniform stream flowing parallel to a long flat plate aligned with the x-axis.Boundary layer thickness d at some location x along the plate is sketchedin Fig. 10–77. By convention, d is usually defined as the distance awayfrom the wall at which the velocity component parallel to the wall is 99 per-cent of the fluid speed outside the boundary layer. It turns out that for agiven fluid and plate, the higher the free-stream speed V, the thinner theboundary layer (Fig. 10–77). In nondimensional terms, we define theReynolds number based on distance x along the wall,

Reynolds number along a flat plate: (10–60)

Hence,

At a given x-location, the higher the Reynolds number, the thinner theboundary layer.

In other words, the higher the Reynolds number, the thinner the boundarylayer, all else being equal, and the more reliable the boundary layer approx-imation. We are confident that the boundary layer is thin when d !! x (or,expressed nondimensionally, d/x !! 1).

The shape of the boundary layer profile can be obtained experimentallyby flow visualization. An example is shown in Fig. 10–78 for a laminarboundary layer on a flat plate. Taken over 50 years ago by F. X. Wortmann,this is now considered a classic photograph of a laminar flat plate boundary

Rex "rVxm

"Vxn

512FLUID MECHANICS

y

d(x)

Rex ~ 102

Vx

y

d(x)

Rex ~ 104

Vx

(a)

(b)

FIGURE 10–77Flow of a uniform stream parallel to a flat plate (drawings not to scale):(a) Rex ! 102, (b) Rex ! 104. Thelarger the Reynolds number, the thinnerthe boundary layer along the plate at agiven x-location.

FIGURE 10–78Flow visualization of a laminar flat plate boundary layer profile.Photograph taken by F. X. Wortmannin 1953 as visualized with the telluriummethod. Flow is from left to right, andthe leading edge of the flat plate is farto the left of the field of view.Wortmann, F. X. 1977 AGARD Conf. Proc. no.224, paper 12.

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Flat Boundary Layer


layer profile. The no-slip condition is clearly verified at the wall, and thesmooth increase in flow speed away from the wall verifies that the flow isindeed laminar.

Note that although we are discussing boundary layers in connection withthe thin region near a solid wall, the boundary layer approximation is notlimited to wall-bounded flow regions. The same equations may be appliedto free shear layers such as jets, wakes, and mixing layers (Fig. 10–79),provided that the Reynolds number is sufficiently high that these regions arethin. The regions of these flow fields with non-negligible viscous forces andfinite vorticity can also be considered to be boundary layers, even though asolid wall boundary may not even be present. Boundary layer thickness d(x)is labeled in each of the sketches in Fig. 10–79. As you can see, by conven-tion d is usually defined based on half of the total thickness of the free shearlayer. We define d as the distance from the centerline to the edge of theboundary layer where the change in speed is 99 percent of the maximumchange in speed from the centerline to the outer flow. Boundary layer thick-ness is not a constant, but varies with downstream distance x. In the exam-ples discussed here (flat plate, jet, wake, and mixing layer), d(x) increaseswith x. There are flow situations however, such as rapidly accelerating outerflow along a wall, in which d(x) decreases with x.

A common misunderstanding among beginning students of fluid mechan-ics is that the curve representing d as a function of x is a streamline of theflow—it is not! In Fig. 10–80 we sketch both streamlines and d(x) for theboundary layer growing on a flat plate. As the boundary layer thicknessgrows downstream, streamlines passing through the boundary layer mustdiverge slightly upward in order to satisfy conservation of mass. Theamount of this upward displacement is smaller than the growth of d(x).Since streamlines cross the curve d(x), d(x) is clearly not a streamline(streamlines cannot cross each other or else mass would not be conserved).

For a laminar boundary layer growing on a flat plate, as in Fig. 10–80,boundary layer thickness d is at most a function of V, x, and fluid propertiesr and m. It is a simple exercise in dimensional analysis to show that d/x is afunction of Rex. In fact, it turns out that d is proportional to the square rootof Rex. You must note, however, that these results are valid only for a lami-nar boundary layer on a flat plate. As we move down the plate to larger andlarger values of x, Rex increases linearly with x. At some point, infinitesimaldisturbances in the flow begin to grow, and the boundary layer cannotremain laminar—it begins a transition process toward turbulent flow. For asmooth flat plate with a uniform free stream, the transition process beginsat a critical Reynolds number, Rex, critical ≅ 1 ! 105 , and continues untilthe boundary layer is fully turbulent at the transition Reynolds number,Rex, transition ≅ 3 ! 106 (Fig. 10–81). The transition process is quite compli-cated, and details are beyond the scope of this text.

513CHAPTER 10

(a)

(b)

(c)

d(x)

x

d(x)

x

V

V

d(x)

x

V2

V1

FIGURE 10–79Three additional flow regions where

the boundary layer approximation maybe appropriate: (a) jets, (b) wakes, and

(c) mixing layers.

V yStreamlines

d(x)d(x)d(x)

Boundary layerx

d(x) FIGURE 10–80Comparison of streamlines and the

curve representing d as a function of xfor a flat plate boundary layer. Sincestreamlines cross the curve d(x), d(x)

cannot itself be a streamline of the flow.

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𝛿 is proportional to the square root of Rex. These results are valid only for a laminar boundary layer on a flat plate.

As we move down the plate to larger and larger values of x, Rex increases linearly with x. At some point, infinitesimal disturbances in the flow begin to grow, and the boundary layer cannot remain laminar—it begins a transition process toward turbulent flow.

For a smooth flat plate with a uniform free stream, the transition process begins at a critical Reynolds number, Rex, critical ≅105, and continues until the boundary layer is fully turbulent at the transition Reynolds number, Rex, transition ≅ 3·106

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Not to scale

To scale

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BL Equations: we restrict attention to steady, 2D, laminar flow (although method is fully applicable to unsteady, 3D, turbulent flow)

BL coordinate systemx : tangential directiony : normal direction

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To derive the equations, start with the steady nondimensional NS equations

Recall definitions

Since , Eu ~ 1Re >> 1, Should we neglect viscous terms? No!, because we would end up with the Euler equation along with deficiencies already discussed.

Can we neglect some of the viscous terms?

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To answer question, we need to redo the nondimensionalization

Use L as length scale in streamwise direction and for derivatives of velocity and pressure with respect to x.

Use δ (boundary layer thickness) for distances and derivatives in y.Use local outer (or edge) velocity Ue.

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Orders of Magnitude (OM)

What about V? Use continuity

Since

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Now, define new nondimensional variables

All are order unity, therefore normalizedApply to x- and y-components of NSE

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Incompressible Laminar Boundary Layer Equations

Continuity

X-Momentum

Y-Momentum

and 10–63, we define the following nondimensional variables within theboundary layer:

Since we used appropriate scales, all these nondimensional variables are oforder unity—i.e., they are normalized variables (Chap. 7).

We now consider the x- and y-components of the Navier–Stokes equation.We substitute these nondimensional variables into the y-momentum equa-tion, giving

u*U

After some algebra and after multiplying each term by L2/(U2d), we get

(10–64)

Comparing terms in Eq. 10–64, the middle term on the right side is clearlyorders of magnitude smaller than any other term since ReL ! UL/n "" 1.For the same reason, the last term on the right is much smaller than the firstterm on the right. Neglecting these two terms leaves the two terms on theleft and the first term on the right. However, since L "" d, the pressure gra-dient term is orders of magnitude greater than the advective terms on theleft side of the equation. Thus, the only term left in Eq. 10–64 is the pres-sure term. Since no other term in the equation can balance that term, wehave no choice but to set it equal to zero. Thus, the nondimensional y-momentum equation reduces to

or, in terms of the physical variables,

Normal pressure gradient through a boundary layer: (10–65)

In words, although pressure may vary along the wall (in the x-direction),there is negligible change in pressure in the direction normal to the wall.This is illustrated in Fig. 10–88. At x ! x1, P ! P1 at all values of y acrossthe boundary layer from the wall to the outer flow. At some other x-location,x ! x2, the pressure may have changed, but P ! P2 at all values of y acrossthat portion of the boundary layer.

The pressure across a boundary layer (y-direction) is nearly constant.

Physically, because the boundary layer is so thin, streamlines within theboundary layer have negligible curvature when observed at the scale of theboundary layer thickness. Curved streamlines require a centripetal accelera-tion, which comes from a pressure gradient along the radius of curvature.Since the streamlines are not significantly curved in a thin boundary layer,there is no significant pressure gradient across the boundary layer.

#P#y

! 0

#P*#y*

! 0

u* #v*#x*

$ v* #v*#y*

! % aLdb 2

#P*#y*

$ a nULb

#2v*#x*2 $ a n

ULb aLdb 2

#2v*#y*2

n #2

#y*2 v*Ud

Ld2n

#2

#x*2 v*Ud

L3

1r

#

#y* P*rU 2

d

#

#y* v*Ud

Ld

v* UdL#

#x* v*Ud

L2

u #v#x

$ v #v#y

! %1r

#P#y

$ n #2v#x 2 $ n

#2v#y 2

x* !xL y* !

y

d u* !

uU v* !

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P* !P % P&

rU 2

517CHAPTER 10

V} F } F V VWall

Boundary layerOuter flow

xx2

x1

P2

P2

P1

P1

P1

P1

P2

P2

y

d

FIGURE 10–88Pressure may change along a

boundary layer (x-direction), but thechange in pressure across a boundary

layer (y-direction) is negligible.

cen72367_ch10.qxd 11/4/04 7:20 PM Page 517

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Boundary Layer Procedure

1. Solve for outer flow, ignoring the BL. Use potential flow (irrotational approximation) or Euler equation

2. Assume δ/L << 1 (thin BL)3. Solve BLE

y = 0 ⇒ no-slip, u=0, v=0

y = δ ⇒ U = Ue(x)

x = x0 ⇒ u = u(x0), v=v(x0)

4. Calculate δ, θ, δ*, τw, Drag

5. Verify δ/L << 1

6. If δ/L is not << 1, use δ* as body and goto step 1 and repeat

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Possible Limitations1. Re is not large enough ⇒ BL may

be too thick for thin BL assumption.

2. ∂p/∂y ≠ 0 due to wall curvature δ ~ R

3. Re too large ⇒ turbulent flow at Re = 1x105. BL approximation still valid, but new terms required.

4. Flow separation

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Before defining and δ* and θ, are there analytical solutions to the BL equations?

Unfortunately, NO

Blasius Similarity Solution boundary layer on a flat plate, constant edge velocity, zero external pressure gradient

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Blasius Similarity Solution

Blasius introduced similarity variables

This reduces the BLE to

This ODE can be solved using Runge-Kutta techniqueResult is a BL profile which holds at every station along the flat plate

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Boundary layer thickness can be computed by assuming that δ corresponds to point where U/Ue = 0.990. At this point, η = 4.91, therefore

Wall shear stress τw and friction coefficient Cf,x can be directly related to Blasius solution

Recall

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Displacement Thickness

Displacement thickness δ* is the imaginary increase in thickness of the wall (or body), as seen by the outer flow, and is due to the effect of a growing BL.

Expression for δ* is based upon control volume analysis of conservation of mass

Blasius profile for laminar BL can be integrated to give

(≈1/3 of δ)

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Momentum Thickness

Momentum thickness θ is another measure of boundary layer thickness.

Defined as the loss of momentum flux per unit width divided by ρU2 due to the presence of the growing BL.Derived using CV analysis.

θ for Blasius solution, identical to Cf,x

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Turbulent Boundary Layer

Illustration of unsteadiness of a turbulent BL

Black lines: instantaneousPink line: time-averaged

Comparison of laminar and turbulent BL profiles

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All BL variables [U(y), δ, δ*, θ] are determined empirically.One common empirical approximation for the time-averaged velocity profile is the one-seventh-power law

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Flat plate zero-pressure-gradient TBL can be plotted in a universal form if a new velocity scale, called the friction velocity Uτ, is used. Sometimes referred to as the “Law of the Wall”

Velocity Profile in Wall Coordinates

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Despite it’s simplicity, the Law of the Wall is the basis for many CFD turbulence models.Spalding (1961) developed a formula which is valid over most of the boundary layer

κ, B are constants

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Pressure Gradients

Shape of the BL is strongly influenced by external pressure gradient:

(a) favorable (dP/dx < 0)

(b) zero

(c) mild adverse (dP/dx > 0)

(d) critical adverse (τw = 0)

(e) large adverse with reverse (or

separated) flow

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Pressure Gradients

The BL approximation is not valid downstream of a separation point because of reverse flow in the separation bubble.Turbulent BL is more resistant to flow separation than laminar BL exposed to the same adverse pressure gradient

Laminar flow separates at corner

Turbulent flow does not separate

me33: fluid flow lecture 1: information and ... - units.it€¦ · downward force: (1) the...

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