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Abstract

Navier-Stokes Equations; Vorticity; Pipe Hagen-Poiseuille Flow; Laminar Flow; Transition; Turbulence; Reynolds Number; Randomness

In this paper, we show that the spatio-temporal evolution of incompressible flows in a long circular pipe can be described by vorticity dynamics. The principal techniques to obtain solution are similar to those used for flows in . As the consideration of the Navier-Stokes equations is given in a cylindrical co-ordinates, two aspects of complication arise. One is the interaction of the velocity components in the radial and azimuthal directions due to the fictitious centrifugal force in the equations of motion. The rate of the vorticity production at pipe wall depends on the initial data at entry and hence is unknown a priori; it must be determined as part of the solution. The vorticity solution obtained defines an intricate flow field of multitudinous degrees of freedom. As the Reynolds number increases, the analytical solution predicts vorticity-scale proliferations in succession. For sufficient large initial data, pipe flows are of turbulent nature. The solution of the governing equations is globally regular and does not bifurcate in space or in time. It is asserted that laminar-turbulent transition is a dynamic process inbred in the non-linearity. The presence of exogenous disturbances, due to imperfect test environments or purpose-made artificial forcing, distorts the course of the intrinsic transition. The flow structures observed by Reynolds (1883) and others can be synthesised and elucidated in light of the current theory.

Navier-Stokes Equations in Cylindrical Co-ordinates]Vorticity evolution in a rigid pipe of circular cross-section F. Lam]F. Lam

Yet every exact solution of the equations of fluid mechanics can actually occur in Nature. The flows must not only obey the equations which are non-linear, but also evolve in the distinct dynamic states as observed.

1 Introduction

In the Eulerian description of the motion of an incompressible, homogeneous Newtonian fluid, the momentum and the continuity equations for fluid dynamics are

(1.1)

where the velocity vector is the velocity, the scalar quantity is the pressure, the space variable is denoted by , and is the Laplacian. The density and the viscosity of the fluid are denoted by and respectively. The kinematic viscosity is . These equations are known as the Navier-Stokes equations (Navier 1823; Stokes 1845). They are derived on the basis of the continuum hypothesis (see, for example, Lamb 1975; Batchelor 1967).

We seek the solution of (1) as an initial-boundary value problem in the space-time domain denoted by , where has a smooth impermeable boundary . The initial condition is given by

(1.2)

and the no-slip boundary condition is

(1.3)

The postulation that is a smooth function with compact support is to simplify the technicalities in the subsequent analysis.

The vector quantity, known as the vorticity , is related to the velocity by . The vorticity field must be solenoidal because of the vector formula . The vorticity is a direct consequence of conservation of the angular momentum of fluid particles. Because of fluid viscosity, particles are subjected to shearing force during motion. Locally, they behave like a rigid body in rotation, at the same time, their moments of inertia redistribute as a result of deformation. The two effects give rise to the dynamic equations,

(1.4)

Because the pressure has been eliminated, the vorticity equations are more accessible in deriving a priori bounds which are critical in global regularity theory of the Navier-Stokes equations (Lam 2013). In particular, it is verified that the total vorticity is an invariant in any fluid motion with smooth initial velocity of bounded energy,

Consequently every vorticity component must be finite. By virtue of the Sobolev embedding theorem, the invariant principle can be extended to show that

(1.5)

where is the small time limit for smoothness in the classical regularity theory. Since the vorticity is a fundamental quantity in fluid motion, we need an inverse relation which enables us to compute the velocity and the pressure. In view of the incompressibility hypothesis and of the vector identity

(1.6)

an elliptic equation can be derived and used to bound in space because di-vorticity, , is a sub-set of Jacobian, . The space-regularity in the velocity gradients ensures the pressure gradient is suitably bounded. From the momentum equation, we assert that

(1.7)

In conjunction with the classical local-in-time regularity, the global well-posedness follows for initial smooth data of finite energy (see, for example, Doering & Gibbon 1995; Constantin 2001; Heywood 2007). By a similarity reduction procedure, it has been shown that, at high Reynolds number, the vorticity field is nothing more than an amalgamation of viscous shearing motions, giving rise to multitudinous spatio-temporal scales. In the sequel, we shall work with infinitely differentiable bounded functions as no finite-time singularity can develop in solution of the Navier-Stokes equations. Viscosity of fluids, no matter how small it may be, activates energy consumption; the non-linearity in the equations facilitates the dissipation by self-multiplying viscous shears into smaller sizes. In fact, the continuity defines a zero-sum of three extended reals, at any given instant of time. None of them can go out of bound as the “measure” , where is a finite real, is undefined in analysis. The physics is clear; any motion due to an incompressible flow of finite energy cannot instigate an unbounded plenum adjacent to an infinity vacuum in any finite region whose flow rate of matter is finite.

The space-time vorticity boundedness, , holds in the limit of vanishing viscosity . Hence the Euler equations () cannot blow up in finite time according to the BKM theorem (Beale et al 1984), assuming that there exists a fluid of zero viscosity. The impossibility of singularities in inviscid flows of finite energy is consistent with Helmholtz’s vortex theorems (1858) and Kelvin’s circulation principle (1869). Owning to lack of energy dissipation, flow motions described by the Euler can only re-distribute their energy. Thus it is comprehensible that, at least locally, inviscid flow fields may evolve into less smooth states compared to their viscous counter-parts. Nevertheless, the Euler equations do not constitute a self-contained model for the physics of turbulence where diffusive dissipation by viscosity is of importance in vorticity proliferation.

2 Velocity-vorticity formulation

In the present paper, we deal with flows in the presence of solid surface and we must strictly impose the no-slip boundary condition (1). We first establish that, in the “kinematic” system

(2.8)

the velocity can be recovered uniquely from

(2.9)

where the region is simply-connected having a boundary . The necessary condition to confirm (2.9) is the vector identity (1.6). The continuity implies that the vorticity field in is solenoidal . Denote by Green’s function satisfying the homogeneous boundary condition. Conversely, if we are given a bounded vector function in . The Poisson equation, , is inverted

(2.10)

where the boundary term from integration by parts vanishes because function is zero on the boundary, and the last integral holds in view of the symmetric property of Green’s function: . Thus the incompressibility is verified if we take divergence operation. Taking curl on the Poisson equation driven by , we obtain

If the vector is solenoidal and coincides with a.e., then the function generally holds such that , and , and this system has no boundary conditions. Given as a vector, we must have or the last elliptic system reduces to

where is a constant. Thus the following condition can be established:

(2.11)

which defines a vortex sheet over the boundary. In application, we have two scenarios: (1) The volume is unbounded. Since we are interested in fluid motions of finite energy, there can be no singular vortex sheets on the boundary in view of (1.5) and (1.7). The function is nothing but the vorticity. (2) For domains of finite , the function can be fixed by initial data (say at ). The problem is to trace the spatio-temporal flow development from a set of initial-boundary conditions by solving the Navier-Stokes equations. In particular, if the initial vorticity in is zero, the system of (2.8) and (2.9) is well-posed in any singly-connected region with smooth boundaries; the velocity can be recovered uniquely from the vorticity (2.10). Our ultimate aim is to solve the dynamic equation (1) or (1) in where both and exist and evolve in real fluids. The use of the dynamic no-slip boundary condition is therefore justified as homogeneous Dirichlet condition (1) is valid at every given instant of time.

In addition, imposing (1) has another advantage which simply fixes the velocity field in the Helmholtz decomposition. Let us write

The function can be determined by taking curl operation so that as long as the -field is solenoidal. The continuity demands that is harmonic

In , it is postulated that the “potential” decays at infinity, or we consider only finite energy solutions as a requirement for physics. Thus a harmonic function bounded in must be a constant by virtue of Liouville’s theorem. The contribution from the irrotational part drops out implicitly. For real fluids in , the no-slip condition adds the boundary condition . The solvability constraint for the Laplace equation,

holds. Hence the only solution for the equation is a constant. Briefly, the formulation (2.8)-(2.9) uniquely specifies the velocity or the vorticity field for real fluids .

3 Dynamic equations in cylindrical co-ordinates

In a cylindrical co-ordinates system , we consider the initial-boundary value of flow evolution. The velocity and pressure . Let denote the velocity components in the radial, circumferential and axial directions respectively. Without loss of generality, we take the pipe radius (denoted by ) as unity. The domain of interest consists of the interior of a semi-infinite circular cylinder

(3.12)

We denote the wall at in . The cross-section at the location is the pipe entry. The pipe surface is assumed to be hydraulically smooth. The choice of the co-ordinates attempts to model, as closely as possible, practical laboratory conditions. In many pipe-flow experiments, controlled water or air from a reservoir is forced into the pipe under external pressure differences. By experience, good flows may be generated in practice by maintaining a constant mass flow through the entire pipe test section while, at the same time, care must be taken to minimise fluctuations in the flow rate. The flow far downstream is inevitably in turbulent stage and the presence of the turbulence needs to be accounted for in theoretical treatment. The axial length is typically - times of pipe diameter, depending on specific set-up. Although the practical pipe length may not be long enough for the flow to decay completely at the far end of the pipe, the flow field close to the inlet must be, to a good approximation, independent of downstream flow.

Momentum equations

The continuity equation reads

(3.13)

The Navier-Stokes momentum equations are

(3.14)

where is the cylindrical Laplacian

(3.15)

and the differential operator , and for a scalar (see, for example, Batchelor 1967; Schlichting 1979; Wu et al 2006).

The pressure satisfies the following Poisson equation:

(3.16)

Hence the pressure can be determined uniquely with respect to a reference pressure once the velocity gradients are known. The elliptic equation contains no time-wise information; any variation in the velocity gradients causes instantaneously changes in the pressure. This apparently unphysical causality effect is a consequence of the continuity (3.13). If the density of the fluid is assumed to vary with the pressure, as given in a state equation, the pressure variation propagates at the local speed of sound. In standard laboratory conditions, the speeds of sound for pure water and air are about m/s and m/s respectively. Thus the incompressibility is a well-suited hypothesis in pipe flow experiments.

We suppose that the motion starts impulsively from rest at time and the flow is initiated afterward. The initial condition is taken as

(3.17)

Equations (3.13) and (3.14) are to be solved subject to the following boundary conditions . The velocity must satisfy the no-slip condition at wall

(3.18)

and decays at infinity

(3.19)

The decay is plausible in physics for flows of finite energy. In addition, the velocity inside the pipe is assumed to be bounded everywhere,

(3.20)

Because of the azimuthal symmetry, it is evident that the velocity must satisfy the periodic condition

(3.21)

at every instant of time. For , we postulate that the flow condition at the pipe entry can be specified as smooth functions of position and time

(3.22)

Since the entry velocity is also governed by the equations of motion, we require that and the initial energy is finite.

Vorticity equations

The vorticity components in the cylindrical co-ordinates are

(3.23)

in the , , directions respectively. In the vector identity, , the quantity on the left-hand side is known as the di-vorticity which links the velocity and the vorticity

(3.24)

for incompressible flows. In terms of the di-vorticity, the Navier-Stokes momentum equations (3.14) can be re-written as

(3.25)

where , is the Bernoulli-Euler pressure (cf. (3.16)). Taking the curl on (3.25) and (3.13), and making use of the vector identity , we obtain the dynamics equations for the vorticity components

(3.26)

where the vector components , and . Note that the continuity equation is satisfied. In addition, the pressure drops out of the vorticity formulation (3.26) so that the number of unknowns is now three, instead of four. The dynamic pressures, and the other derivatives, must be found, indirectly, from one of the momentum equations in (3.14).

As our motion starts impulsively from rest, condition (3.17) means the initial data for vorticity are identically zero

(3.27)

At any subsequent instant during the evolution, the vorticity satisfies the periodic condition

(3.28)

From the velocity distribution (3.22), the vorticity at the entry can be calculated

(3.29)

where we have explicitly expressed or as a function of time. The axial gradient of the entry vorticity is given by

(3.30)

The left-hand sides in (3.26) are diffusion operators. During the vorticity evolution, the pipe wall acts as a source for vorticity in view of the no-slip condition. However, the amount of the vorticity generated across the wall per unit area per unit time is not known and clearly depends on the dynamics of the whole flow field. Theoretically, the wall vorticity must constituent part of the solution. For convenience, we designate the vorticity boundary value as

(3.31)

To simplify our notations, the following short-hands are helpful:

where the middle integral is over the entire pipe wall and it is used to specify the wall vorticity while the last over the inner cross-sectional area. Similarly, space-time integrals are

In our presentation, we use the following notations for the independent variables

Use of them becomes clear when the domain of integration is stated.

In the present paper, we intend to develop an analytical theory for the flow-field evolution from the commencement of motion, given the prescribed data. The fluid dynamics in circular pipe has been subjected to intensive investigations for more than years, starting with Reynolds’ experiments (Reynolds 1883). We refer the reader to a recent review and the references therein on the latest development (Mullin 2011). It is an experimental fact that pipe flow is too mysterious to comprehend. Little may be extracted from experiments as guides for theoretical treatment. It would be fair to acknowledge that laboratory experiments using various exploitation techniques are almost exhaustive as far as incompressible flows are concerned. It is lack of a rigorous mathematical framework which hinders progress because of the difficulties to reconstruct the non-linear characters of fluid dynamics from test data.

4 Kinematics of vorticity induction

The components of Poisson’s equation in (3.24) have the explicit expressions

(4.32)

which are viewed as a system of elliptic equations subject to conditions (3.18) on pipe wall and (3.22) at pipe entry. Green’s function for the Laplacian can be found by the method of separation of variables. In view of the no-slip condition at the wall, it has the form of

(4.33)

where

and the constant, is the th positive zero of Bessel function , the th positive zero of . The zeros, , are larger than (see appendix A for some examples). Similarly, Green’s function for operator is found to be

(4.34)

There exists a rich collection of literature on Bessel functions, see, for example, Watson (1944) and Olver et al (2010). Bessel functions are entire functions of the argument ,

(4.35)

and

(4.36)

The following recurrence relations are well-known (for )

(4.37)

The denominators in the Green functions are derived from the orthogonal relation

where is Kronecker’s symbol. For the Bessel functions of all order ,

For small arguments, we have

The Bessel functions oscillate for large arguments

By the series expansion for the Bessel function, we find

(4.38)

We have carried out some numerical experiments for the double sum

For and , all the computations can be done efficiently. There have been no numerical difficulties.

The inversion of the last equation in (4.32) is given by

(4.39)

where the functions,

(4.40)

are obtained from integration by parts. The boundary terms vanish in view of the periodic condition for vorticity (3.28), and the properties of the Bessel functions. Moreover, the notation , and we have introduced the notation,

to account for the contribution from the velocity at the pipe entry. To continue the process of inversion, we reduce the first two equations in (4.32) to

(4.41)

where the derivatives on Green’s function are derived in the same manner as those in (4.40), and use is made of the velocity periodic condition (3.21), and the decay of Green’s function at . In particular, the function remains regular as by virtue of (4.35) and (4.38). The induced velocities by the entry, and , are found in an analogous manner to that by . In (4.39), we replace the kernel and replaced by and to obtain ,

(4.42)

where . The component is obtained analogously. In view of the incompressibility, the induced velocity is completely determined and fixed for given geometry. It is evident that the influence of the entry data of finite energy is restricted to a portion of the pipe downstream of the entry and diminishes at large distance as both and .

Because of the rotational symmetry from to , we readily verify that

From the properties of the zeros for , neither nor is an eigenvalue of the homogeneous operator . The equations in (4.41) are a pair of Fredholm integral equations of the second kind with continuous bounded kernels. They may be re-written in vector form for the unknown vector

(4.43)

where , and the ad hoc operator stands for integrals of matrix kernel multiplying a vector function. (It should not be confused with the usual convolution operator.) The domain of integration is clear by considering the kernel and the function; it is over the space . Explicitly, the kernel is a non-zero square matrix with zero trace,

(4.44)

and is ,

(4.45)

Pre-multiply (4.43) by , we obtain

Carrying out the multiplication once more, we get

After -times multiplications, we write the result as

(4.46)

As shown earlier, the boundedness for implies . Since we are interested in fluid motions of finite energy, we consider , then the tail on the right in (4.46) vanishes as

where is the identity matrix. By the Sobolev embedding theorem, we may make reference to the -theory for Fredholm integral equations of the second kind (see, for example, Chapter 2 of Tricomi 1957). The operation sum must converge in the limit of . Let us denote the infinite sum of the multiple operations by , and it is called the resolvent kernel of . For bounded vorticity, the solution of the integral equations (4.43) or (4.41) is given by

Combining the result with (4.39), we write the relation between the velocity and the vorticity as , or

(4.47)

This relation holds for every instant of time but it does not contain time information since equation (4.32) is elliptic. This is a consequence of the incompressibility hypothesis (3.13). We draw our attention to the fact that the vortices, and , do contribute to their velocities and in contrast to the velocity induction in the Cartesian co-ordinates. The reason is that Laplacian is linked to the azimuthal gradient in , which is partly driven by vorticity component . When a unit-mass fluid particle at from the pipe centre is rotating in the plane instantaneously, the term in the -momentum equation in (3.25) is the centrifugal acceleration on the particle toward the centre.

It may have been a generalisation of the potential theory or the electromagnetism in classical physics, suggestion has been put forward to represent the velocity field by a solenoidal vector stream function, where . Instead of (4.32), we have

(4.48)

Although this set of lower-order equations appears to be simpler, its general use can be trick and hence is not recommended except for flows in unbounded space . The reason is that no quantitative knowledge of the stream function on solid walls is available; the inversion of (4.48) cannot be readily obtained. The integral of contains arbitrary constants, depending on local flow condition. On the other hand, imposing the no-slip condition involves two Green’s functions for every component of and hence the evaluation of these Green functions becomes substantially more complicated.

5 Vorticity due to entry flow

Because of the specific characters of our co-ordinates, we prefer to write Green’s function for the diffusion kernels in (3.26) as

(5.49)

where is diagonal, and we denote its components in a row matrix

The function,

(5.50)

is the gradient of the fundamental solution of heat equation in one space dimension (). The second part of the Green function satisfying the Neumann homogeneous condition on the wall is written in the diagonal matrix

The first two elements are , where

(5.51)

and

(5.52)

The constant, is the th positive zero of Bessel function , and the th positive zero of . The zeros, , are larger than (see appendix A).

As a first approximation, we investigate the vorticity field caused by the entry vorticity (3.27) and, for the time being, we ignore the contributions from the wall. Then the dynamics equations (3.26) can be expressed in integral form:

(5.53)

where is a diagonal matrix. We denote the elements by

and

where

As implied in , the last integral term represents the non-linear interaction in the vorticity field.

By contrast with the vorticity dynamics in the Cartesian co-ordinates, two extra terms arise in (3.26), namely,

We expect that they are invariant with respect to the rotational symmetry. For instance, we obtain the following result by integration by parts: