. A term which is proportional to the volume dilation∇𝑇𝑣, i.e. the divergence of velocity.
. The part which is proportional to the rate of change of strain in the respective coordinate direction, e.g.𝜕𝑢∕𝜕𝑥.
By separating the pressure from the remaining normal stress, the limiting case for vanishing velocity lets the equations collapse to the hydrostatic case.
A more elaborate discussion about the analogy of the stress–rate of change of strain relationship for fluids and the stress–strain relationship of elastic bodies can be found in Schlichting and Gersten (, Chapter ).
Two material properties appear in the Equations (.) for the stresses: Stokes’
hypothesis
• the dynamic viscosity𝜇and
• the second viscosity𝜆.
Determining these properties is a task of the scientific field of rheology.
In principle, the viscosity can depend on time, position, as well as the state of a flow field itself (temperature, stress, velocity, etc.). Luckily for us, air and water belong to the class of Newtonian fluids. For Newtonian fluids, the material properties𝜇and𝜆 are independent of the actual stress and velocity values.
Dynamic viscosity differs from fluid to fluid. Viscosities of water and air are a function of temperature (see Section ..). Very little is known about the second viscosity𝜆. Since it is associated with changes in volume, it cannot be observed with incompressible fluids. Remember that the divergence of the velocity vanishes for fluids with constant density:∇𝑇𝑣= 0.
Stokes solved the difficulty posed by the unknown second viscosity by drawing an analogy to elastic bodies. He postulated the relationship3𝜆+ 2𝜇= 0between dynamic and second viscosity, which has been named in his honor:
Stokes’ hypothesis for second viscosity 𝜆 = −2
3𝜇 (.)
Our experience shows that the resulting Navier-Stokes equations describe viscous fluid flow correctly over a wide range of applications. However, the validity of Stokes’
hypothesis is still debated among experts. See for instance Gad-el Hak ().
6.4 Navier-Stokes Equations for a Newtonian Fluid
Finally, substituting the stress–rate of change of strain relationships (.) together Navier-Stokes equations
with Stokes’ hypothesis (.) into the conservation of momentum equations (.), (.), and (.) yields the Navier-Stokes equations (NSE) for an isotropic Newtonian
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68 6 Navier-Stokes Equations
fluid in their conservative, differential form, which describe viscous flows.
𝜕( 𝜌𝑢)
𝜕𝑡 + [𝜕(
𝜌 𝑢 𝑢)
𝜕𝑥 + 𝜕( 𝜌 𝑣 𝑢)
𝜕𝑦 + 𝜕( 𝜌 𝑤 𝑢)
𝜕𝑧 ]
= 𝜌𝑓𝑥− 𝜕𝑝
𝜕𝑥
− 𝜕
𝜕𝑥 [2
3𝜇∇𝑇𝑣− 2𝜇𝜕𝑢
𝜕𝑥 ]
+ 𝜕
𝜕𝑦 [
𝜇 (𝜕𝑣
𝜕𝑥+𝜕𝑢
𝜕𝑦 )]
+ 𝜕
𝜕𝑧 [
𝜇 (𝜕𝑢
𝜕𝑧+𝜕𝑤
𝜕𝑥
)] (.)
𝜕( 𝜌𝑣)
𝜕𝑡 + [𝜕(
𝜌 𝑢 𝑣)
𝜕𝑥 + 𝜕( 𝜌 𝑣 𝑣)
𝜕𝑦 + 𝜕( 𝜌 𝑤 𝑣)
𝜕𝑧 ]
= 𝜌𝑓𝑦−𝜕𝑝
𝜕𝑦 + 𝜕
𝜕𝑥 [
𝜇 (𝜕𝑣
𝜕𝑥+ 𝜕𝑢
𝜕𝑦 )]
− 𝜕
𝜕𝑦 [2
3𝜇∇𝑇𝑣− 2𝜇𝜕𝑣
𝜕𝑦 ]
+ 𝜕
𝜕𝑧 [
𝜇 (𝜕𝑤
𝜕𝑦 +𝜕𝑣
𝜕𝑧 )]
(.)
𝜕( 𝜌𝑤)
𝜕𝑡 + [𝜕(
𝜌 𝑢 𝑤)
𝜕𝑥 + 𝜕( 𝜌 𝑣 𝑤)
𝜕𝑦 + 𝜕( 𝜌 𝑤 𝑤)
𝜕𝑧 ]
= 𝜌𝑓𝑧− 𝜕𝑝
𝜕𝑧 + 𝜕
𝜕𝑥 [
𝜇 (𝜕𝑢
𝜕𝑧+ 𝜕𝑤
𝜕𝑥 )]
+ 𝜕
𝜕𝑦 [
𝜇 (𝜕𝑤
𝜕𝑦 + 𝜕𝑣
𝜕𝑧 )]
− 𝜕
𝜕𝑧 [2
3𝜇∇𝑇𝑣− 2𝜇𝜕𝑤
𝜕𝑧
] (.)
All terms in equations (.), (.), and (.) represent forces per unit fluid volume.
Figure . identifies the force components in the NSE. On the left-hand side we have the inertia force and on the right-hand side we see the sum of external forces which comprises body and surface forces.
The Navier-Stokes equations are named after Stokes and Navier, who published a form
C.L. Navier, G.G.
Stokes of the NSE earlier than Stokes (Navier, ). However, Navier had no physical concept for the material parameter we now know as viscosity (Navier, ).
The Navier-Stokes equations are a set of coupled, nonlinear partial differential equations.
Mathematical
classification They are nonlinear because of the convection terms in brackets on the left side of (.), (.), and (.), which contain products of the unknown velocity components. If you also consider the density and viscosity distributions as unknown, then more terms are nonlinear. Luckily, we may assume that kinematic viscosity and density are constant for most problems in ship hydrodynamics. We will exploit this in the next chapter to simplify the equations.
Even today we know very little about the existence and uniqueness of solutions to the Navier-Stokes equations. The Clay Mathematical Institute (CMI) in Cambridge, MA made the Navier-Stokes equations one of their famous Millenium Problems. A contribution which representssubstantial progress toward a mathematical theory which will unlock the secrets hidden in the Navier-Stokes equationscan earn you a $ million award (see http://www.claymath.org/millennium-problems/navier-stokes-equation).
All modern approaches of CFD in ship hydrodynamics actually do not solve the equa-
NSE solvers
tions above but a simplified version of them called the Reynolds averaged Navier-Stokes equations or short RANSE (see Chapter ). The key is that the time dependent flow velocity is subdivided into a time dependent average velocity and a highly fluctuating turbulence. This leads to an additional term in the equations. It has the form of a stress
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6.4 Navier-Stokes Equations for a Newtonian Fluid 69
Figure 6.3 Forces comprising the Navier-Stokes equations for an isotropic Newtonian fluid
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70 6 Navier-Stokes Equations
tensor and in honor of Osborne Reynolds is named the Reynolds stress tensor. For the newly introduced unknowns additional equations are needed, which are known as turbulence models.
References
Anderson, Jr., J. ().Solutions manual to accompany computational fluid dynamics – The basics with applications. McGraw-Hill, Inc., New York, NY.
Gad-el Hak, M. (). Stokes’ hypothesis for a Newtonian, isotropic fluid. Journal of Fluids Engineering, ():–.
Navier, C. (). Mémoire sur les lois du mouvement des fluides.Mém. Acad. Sci. Inst.
France, :–.
Schlichting, H. and Gersten, K. ().Boundary-layer theory. Springer-Verlag, Berlin, Heidelberg, New York, eigth edition. Corrected printing.
Stokes, G. (). On the theories of the internal friction of fluids in motion, and of the equilibrium and motion of elastic solids.Transactions of the Cambridge Philosophical Society, :–.
Self Study Problems
. Draw duplicates of the fluid elementd𝑉 in Figure . and sketch in momentum fluxes for𝑦and𝑧directions. Follow the steps of Equations (.) through (.) and derive the momentum flux in𝑦- and𝑧-direction. You should arrive at the result shown in Equations (.) and (.).
. Summarize in your own words Stokes’ hypothesis.
. Why are we not able to determine the second viscosity𝜆for water?
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71
7
Special Cases of the Navier-Stokes Equations
The Navier-Stokes equations developed in the previous chapter contain the fluid prop- erties density𝜌and dynamic viscosity𝜇. The density is also part of the continuity equation (.). We will simplify both the continuity equation and the NSE for the ship resistance and propulsion problem by making reasonable assumptions about density and viscosity. A dimensionless form of the NSE will provide insight into the different scales of the forces driving the flow.
Learning Objectives
At the end of this chapter students will be able to:
• find the effects of constant density on conservation of mass and momentum
• identify the terms in the NSE as forces per unit mass
• formulate a dimensionless version of the NSE
• introduce characteristic, dimensionless parameters
• explain the challenge of high Reynolds numbers