Dimensionless Navier-Stokes Equations

We perform the same transformations for the𝑦- and𝑧-components of the NSE:

𝜕𝑣

𝜕𝑡 + ( 𝑣𝑇∇)

𝑣 = 𝑓𝑦 − 1 𝜌

𝜕𝑝

𝜕𝑦 + 𝜈𝚫𝑣 (.)

𝜕𝑤

𝜕𝑡 + ( 𝑣𝑇∇)

𝑤 = 𝑓𝑧 − 1 𝜌

𝜕𝑝

𝜕𝑧 + 𝜈𝚫𝑤 (.)

The three components of the NSE may now be assembled into a convenient vector Vector form of NSE

(𝜌, 𝜈= const.)

equation:

𝜕𝑣

𝜕𝑡 + ( 𝑣𝑇∇)

𝑣

⏟⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏟

= 𝑓

⏟⏟⏟

− 1 𝜌∇𝑝

⏟⏟⏟

+ 𝜈𝚫𝑣

⏟⏟⏟

(.)

This looks a lot more compact than the Navier-Stokes equations we developed first in Chapter . The individual terms still represent the same forces as in Figure .. This time, however, all forces are given as accelerations, i.e. force per unit mass.

. Inertia force on the left-hand side.

. Vector of body forces 𝑓.

. Pressure force 1𝜌∇𝑝.

. Viscous force 𝜈𝚫𝑣.

Equation (.) represents the conservation of momentum for an incompressible New- tonian fluid at constant temperature. Together with the continuity equation (.), it describes the motion of water around ships. We now have four equations – the continuity equation and the three components of the NSE – which in principle allow us to find the four unknowns: pressure𝑝and the components𝑢,𝑣, and𝑤of the velocity vector.

In practice, most available CFD systems do not attempt to solve Equation (.) directly.

For high Reynolds numbers this direct numerical simulation (DNS) is still beyond our computational capabilities. Instead, we solve the RANSE, which are derived by splitting the instantaneous time dependent velocity into an average velocity (still time dependent) and the turbulent velocity. This introduces the initially unknown Reynolds stress tensor.

The added unknowns require additional equations to represent the turbulence, so- called turbulence models. The latter are still a focus of ongoing research and several turbulence models of varying complexity are in use for ship resistance computations.

The interested reader can find a derivation of the RANSE form of the Navier-Stokes equations for incompressible flow in Chapter .

7.2 Dimensionless Navier-Stokes Equations

Additional insight into the Navier-Stokes equations can be gained by rewriting them in a dimensionless form. This is general practice for implementation of numerical solutions methods. Therefore, this example might suffice to show this useful technique.

Trim Size: mm×mm Single Column Tight Birk — “fshy” — // — : — page  — #

j j

76 7 Special Cases of the Navier-Stokes Equations

We revisit the NSE (.):

𝜕𝑣

𝜕𝑡 + ( 𝑣𝑇∇)

𝑣 = 𝑓 − 1

𝜌∇𝑝 + 𝜈𝚫𝑣 (.)

As mentioned, both sides represent accelerations and are measured in the physical unit m/s2.

We make quantities dimensionless by dividing them by a suitable reference value. For

Dimensionless

lengths and∇ instance, all space variables are divided by a reference length𝐿: 𝑥∗ = 𝑥

𝐿 𝑦∗ = 𝑦

𝐿 𝑧∗ = 𝑧

𝐿 (.)

This seems fairly obvious. Less apparent is that the Nabla operator has a dimension as well. The dimensionless Nabla operator∇∗is equal to

∇∗ =

⎛⎜

⎜⎜

⎝

𝜕

𝜕𝑥∗

𝜕

𝜕𝑦∗

𝜕

𝜕𝑧∗

⎞⎟

⎟⎟

⎠

⎛⎜

⎜⎜

⎝

𝜕

𝜕 (𝑥

𝐿 )

𝜕

𝜕 (𝑦

𝐿 )

𝜕

𝜕 (𝑧

𝐿 )

⎞⎟

⎟⎟

⎠

⎛⎜

⎜⎜

⎝ 𝐿𝜕

𝜕𝑥 𝐿𝜕

𝜕𝑦 𝐿𝜕

𝜕𝑧

⎞⎟

⎟⎟

⎠

= 𝐿∇ (.)

The dimensionless Laplace operator is derived from the identity𝚫= ∇𝑇∇:

Dimensionless Laplace operator

𝚫 𝚫∗ = (

∇∗)𝑇

∇∗ = ( 𝐿∇)𝑇

𝐿∇ = 𝐿2𝚫 (.)

Any geometric characteristic of the flow problem may serve as reference length𝐿, for example the length or beam of a vessel, the diameter of a propeller, the waterdepth, and others.

We also introduce a reference period𝑇. Possible choices are the duration of a process

Dimensionless

time or the period of a recurring event. Our dimensionless time𝑡∗and partial derivative will be

𝑡∗ = 𝑡 𝑇

𝜕

𝜕𝑡∗ = 𝑇 𝜕

𝜕𝑡 (.)

For convenience, we will select a reference velocity of magnitude𝑈∞, e.g. ship or

Dimensionless

velocity flow speed. In practice, reference length and reference period can be used to define a reference velocity or length and velocity constitute a reference time. The dimensionless velocity vector becomes

𝑣∗ =

⎛⎜

⎜⎜

⎜⎝ 𝑢∗ 𝑣∗ 𝑤∗

⎞⎟

⎟⎟

⎟⎠

⎛⎜

⎜⎜

⎜⎝ 𝑢 𝑈∞

𝑣 𝑈∞

𝑤 𝑈∞

⎞⎟

⎟⎟

⎟⎠

= 1 𝑈∞

⎛⎜

⎜⎜

⎜⎝ 𝑢 𝑣 𝑤

⎞⎟

⎟⎟

⎟⎠

= 𝑣

𝑈∞ (.)

Trim Size: mm×mm Single Column Tight Birk — “fshy” — // — : — page  — #

j j

7.2 Dimensionless Navier-Stokes Equations 77

Finally, we select a reference acceleration for the vector of body forces (per unit mass) Dimensionless gravitation and pressure

𝑓. The gravity force is the only body force of note we will consider. Therefore, we use the gravitational acceleration𝑔= 9.807m/s2as a reference value. For the pressure we select𝑝∞, which often represents the pressure far away from the body.

𝑓∗ = 𝑓

𝑔 𝑝∗ = 𝑝

𝑝∞ (.)

The NSE (.) become dimensionless by replacing the dimensional quantities with Dimensionless

their dimensionless counterparts and reference quantities, i.e.𝑥 = 𝐿𝑥∗,𝑣= 𝑈∞𝑣∗, NSE

𝑝=𝑝∞𝑝∗, and so on.

1 𝑇

𝜕( 𝑈∞𝑣∗)

𝜕𝑡∗ + [(

𝑈∞𝑣∗)𝑇(∇∗ 𝐿

)] (𝑈∞𝑣∗)

= 𝑔𝑓∗ − ∇∗ 𝜌 𝐿

(𝑝∞𝑝∗) + 𝜈

(𝚫∗ 𝐿2

) (𝑈∞𝑣∗) (.) We factor out the reference quantities in each term:

𝑈∞ 𝑇

𝜕𝑣∗

𝜕𝑡∗ + 𝑈∞2 𝐿

(𝑣∗𝑇∇∗)

𝑣∗ = 𝑔𝑓∗ − 𝑝∞ 𝜌 𝐿

(∇∗𝑝∗)

+ 𝜈𝑈∞ 𝐿2

(𝚫∗𝑣∗)

(.) Finally, this equation is multiplied by𝐿∕𝑈∞2 and we obtain:

[ 𝐿 𝑇 𝑈∞

]𝜕𝑣∗

𝜕𝑡∗ +( 𝑣∗𝑇∇∗)

𝑣∗ = [

𝑔𝐿 𝑈∞2

] 𝑓∗ −

[ 𝑝∞ 𝜌 𝑈∞2

]

∇∗𝑝∗ + [ 𝜈

𝐿𝑈∞ ]

𝚫∗𝑣∗ (.)

All factors in (.) are now dimensionless, including the four terms in brackets which Characteristic numbers of the

are important numbers characterizing the flow around a ship hull. flow

. On the left-hand side we have a form of the Strouhal number Strouhal number

𝑆𝑡 = 𝐿

𝑇 𝑈∞ Strouhal number (.)

Vincenz Strouhal (*–†) was a Czech physicist who introduced this dimensionless number in  (Strouhal, ).

You probably have heard taught wires (rigging, guides of a cell tower) hum in high winds. The sound is created when the periodic shedding of vortices in the flow causes the wire to vibrate. The Strouhal number𝑆𝑡is used to describe the frequency of unsteady processes like vortex shedding. Therefore, it precedes the local time derivative of the velocity𝜕𝑣∗∕𝜕𝑡∗, which represents the acceleration in the dimensionless Navier-Stokes equations (.).

. The first dimensionless number on the right-hand side is connected to the Froude Froude number

number𝐹𝑟.

[ 𝑔𝐿 𝑈∞2

]

= 1

𝐹𝑟2 with 𝐹𝑟 = 𝑈∞

√𝑔𝐿 Froude number (.)

Trim Size: mm×mm Single Column Tight Birk — “fshy” — // — : — page  — #

j j

78 7 Special Cases of the Navier-Stokes Equations

The Froude number𝐹𝑟is named after William Froude, the father of modern ship model testing. Naval architects use the Froude number as a dimensionless velocity. It is connected to the gravity forces and important for the similarity of wave patterns.

. The next pair of brackets encloses a form of Euler number named after Leonard

Euler number

Euler.

𝐸𝑢 = [

𝑝∞ 𝜌 𝑈∞2

]

Euler number of fluid mechanics (.) Euler made many important contributions to mathematics, structural mechanics, and fluid mechanics, among others, and various theories, equations, as well as numbers bear his name. In practice, we are more likely to encounter dimensionless pressure coefficients𝐶𝑝 which compare pressure differences with the dynamic pressure(1∕2)𝜌 𝑈∞2:

𝐶𝑝 = 𝑝−𝑝∞

2𝜌 𝑈∞2 pressure coefficient (.)

. The last dimensionless number in brackets is the reciprocal of the Reynolds

Reynolds

number number𝑅𝑒:

[ 𝜈 𝐿𝑈∞

]

= 1

𝑅𝑒 with 𝑅𝑒 = 𝐿𝑈∞

𝜈 Reynolds number (.)

The Reynolds number is named after Osborne Reynolds, who studied, among many other things, the flow in pipes and under which conditions the flow tran- sitioned from laminar to turbulent. The dimensionless Reynolds number is obviously connected to the viscous forces caused by the viscosity of the water.

Froude number and Reynolds number are arguably the most important qualifiers of flow conditions in ship hydrodynamics. We will encounter these numbers many times in subsequent chapters.

We now substitute the dimensionless numbers𝑆𝑡,𝐹𝑟,𝐸𝑢, and𝑅𝑒for the square brackets

Dimensionless

NSE into the Navier-Stokes equations:

𝑆𝑡𝜕𝑣∗

𝜕𝑡∗ + ( 𝑣∗𝑇∇∗)

𝑣∗ = 1

𝐹𝑟2𝑓∗ − 𝐸𝑢∇∗𝑝∗ + 1

𝑅𝑒𝚫∗𝑣∗ (.) An example will teach us something about the terms in the Navier-Stokes equations.

Example

Consider the following data set:

length of vessel 𝐿𝑊 𝐿= 120.00 m

ship speed 𝑣𝑠= 20.00 kn

period 𝑇 = 6.59 s

gravitational acceleration 𝑔= 9.81 m/s2

density of salt water at 𝑜C 𝜌𝑠= 1026.021kg/m3 standard atmospheric pressure 𝑝∞= 101325.00 Pa kinematic viscosity of seawater at 𝑜C 𝜈= .⋅10−6 m2/s

Trim Size: mm×mm Single Column Tight Birk — “fshy” — // — : — page  — #

j j

7.2 Dimensionless Navier-Stokes Equations 79

We will employ the numbers to calculate the four dimensionless constants of the Navier- Stokes equations.

. Strouhal number – Before we substitute the values above into Equation (.), we must convert the ship speed into SI units. One knot is equal to one nautical mile per hour, i.e. kn=M/h and

1kn = 1852m/M

3600s/h = 0.51444m h/(s M) (.) Therefore, the ship speed we want to use as a reference velocity is

𝑣𝑠 = 20kn = 10.289m/s = 𝑈∞

As a reference period, we use the period of the transverse waves generated by the ship. We will discuss this in depth later. With the reference length𝐿= 120m, reference period𝑇 = 6.59s, and reference velocity𝑈∞= 10.289m/s we obtain a Strouhal number of

𝑆𝑡 = 𝐿

𝑇 𝑈∞ = 120m

6.59s10.289m/s = 1.76984 It is good practice to check that the result is truly dimensionless.

. Froude number – With the preparation above, the Froude number poses no challenge:

𝐹𝑟 = 𝑈∞

√𝑔𝐿

= 10.289m/s

√

9.81m/s2⋅120m = 0.29988

Please note, that the Froude number is usually computed on the basis of length in waterline𝐿𝑊 𝐿.

The coefficient for the Navier-Stokes equations is 1

𝐹𝑟2 = 0.29988−2 = 11.12

. Euler number – In this example we get for the Euler number 𝐸𝑢 = 𝑝∞

𝜌𝑈∞2 = 101325kg/(ms2)

1026.021kg/m3⋅10.2892(m/s)2 = 0.93288

. Reynolds number – The Reynolds number is:

𝑅𝑒 = 𝐿𝑈∞

𝜈 = 120m⋅10.289m/s

1.1892⋅10−6m2/s = 1.03823⋅109

Note that the Reynolds number is usually based on the length over wetted surface 𝐿𝑂𝑆.

The coefficient for the Navier-Stokes equations is the inverse of the Reynolds number

𝑅𝑒 = 1

1.03823⋅109 = 9.632⋅10−10

Trim Size: mm×mm Single Column Tight Birk — “fshy” — // — : — page  — #

j j

80 7 Special Cases of the Navier-Stokes Equations

We rewrite the Navier-Stokes equations using the coefficients we computed for this example:

1.76984𝜕𝑣∗

𝜕𝑡∗ + ( 𝑣∗𝑇∇∗)

𝑣∗

= 11.12𝑓∗ − 0.93288 ∇∗𝑝∗ + 9.632⋅10−10𝚫∗𝑣∗ (.) Three of the coefficients are in the range of1, but the coefficient for the body forces is of magnitude10, and the coefficient for the viscous forces is very small, roughly10−9. Thus, the factor for the body forces is10 000 000 000times larger than the factor for the viscous forces! This seems to indicate that viscous forces may be negligible.

Indeed, there are flow phenomena which support this notion. We know from detailed observations that waves created by storms in the vicinity of the Antarctic Circle travel halfway around the world to hit the shores of the North American Pacific coast. Their height diminishes somewhat along the way but more due to adverse winds than friction.

On the other hand, it is a well known fact that if you give a boat a push, it will come to rest again after a short while. Even if we avoid the generation of waves by considering a well streamlined, submerged body, it will come to rest fairly soon without a continuous propulsive force.

This apparent discrepancy between theory, in the form of the coefficients in the Navier-

Boundary layer

theory Stokes equations, and the reality of observations vexed scientists and engineers at the end of the th century. It was Ludwig Prandtl who proposed in  to divide the flow around bodies into two regions: a thin sheet of fluid close to the body surface, called the boundary layer, where viscous effects are present and an exterior flow outside the boundary layer where viscous effects are mostly negligible. Prandtl introduced the concept of fluid molecules sticking to the surface and also offered an explanation for the phenomenon of flow separation. More on this later. Readers are encouraged to read the paper by Anderson, Jr. () on Prandtl’s boundary layer theory and its impact on aerodynamics and fluid mechanics.

Prandtl’s ideas allow a simplification of the Navier-Stokes equations into the boundary layer equations. Although the boundary layer equations are a special case of the Navier- Stokes equations, we will discuss them in subsequent chapters. They form the basis of skin friction computations and provide important insights into the flow around ship hulls.

References

Anderson, Jr., J. (). Ludwig Prandtl’s boundary layer.Physics Today, ():–.

Strouhal, V. (). ĩber eine besondere Art der Tonerregung. NF. Bd. V():–.

Deutsches Textarchiv http://www.deutschestextarchiv.de/strouhal_tonerregung_

, last visited July , .

Trim Size: mm×mm Single Column Tight Birk — “fshy” — // — : — page  — #

j j

7.2 Dimensionless Navier-Stokes Equations 81

Self Study Problems

. Look at the following equation of fluid mechanics:

𝜕𝑣

𝜕𝑡 + ( 𝑣𝑇∇)

𝑣

⏟⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏟

= 𝑓

⏟⏟⏟

− 1 𝜌∇𝑝

⏟⏟⏟

+ 𝜈𝚫𝑣

⏟⏟⏟

4 Answer the following questions:

(a) What type of equation is this mathematically?

(b) What is the name of the equation?

(d) What is the dimension of the terms in the equation?

. Provide a definition for each of the four dimensionless numbers:𝑅𝑒,𝐹𝑟,𝐸𝑢,𝑆𝑡.

. Consider the following form of the continuity equation.

𝜕𝜌

𝜕𝑡 + ∇𝑇(𝜌𝑣) = 0

(a) State whether the equation is in differential or integral form and whether it is the conservative or nonconservative form.

(b) Convert the equation into a dimensionless form based on reference quanti- ties𝐿for length,𝑇 for time,𝑈∞for velocity, and𝜌∞for density.

Trim Size: mm×mm Single Column Tight Birk — “fshy” — // — : — page  — #

j j

Reynolds Averaged Navier-Stokes Equations (RANSE)

In practice, most available CFD systems do not attempt to solve Equation (.) directly. For high Reynolds numbers direct numerical simulation is still beyond our computational capabilities. Instead, we solve the RANS equations, which are derived by splitting the instantaneous time dependent velocity into an average velocity (still time dependent) and the turbulent velocity. This introduces the initially unknown Reynolds stress tensor. The added unknowns require additional equations to represent the turbulence. Turbulence modeling is still a focus of ongoing research, and several turbulence models of varying complexity are in use for ship resistance computations.

We derive the RANSE form of the Navier-Stokes equations for incompressible flow below to illustrate the difference.

Learning Objectives

At the end of this chapter students will be able to:

• transform the NSE into RANSE

• formulate a dimensionless version of the NSE

• introduce characteristic dimensionless parameters

• explain the challenge of high Reynolds numbers

Navier-Stokes Equations for a Newtonian Fluid