**6. Conclusions**

The method discussed in this chapter eliminates the convection term in Eq. (1) and makes it feasible to use central difference schemes to solve convection-diffusion equations accurately. The new method, combined with the central difference schemes, can achieve better accuracy than the upwind schemes on the same finite difference stencil, and is shown in the examples presented here to be as robust as the upwind schemes for convection dominated problems. It can also be easily combined with the Padé approximation to achieve fourth-order accuracy in space on a 3-point finite difference stencil.

The new method does incur a modest increase in computational complexity. Instead of just solving Eq. (1), the new method requires solving Eqs. (5) and (7). With a 3-point stencil, the standard upwind-central difference schemes will generate a system of tri-diagonal algebraic equations, while the new method discussed in this chapter will lead to a system of block tri-diagonal algebraic equations with 2 × 2 blocks. This increased complexity, however, can be compensated by the use of fewer grid points with the increased order of accuracy of the new method. Furthermore, for problems that require calculations of both the solutions and their derivatives, the new method eliminates the need to calculate the derivatives after solving Eq. (1).

The discussions of the new method in this chapter are based on one-dimensional problems. For higher-dimensional problems, a straightforward application of this method will lead to systems of four equations for two-dimensional problems and seven equations for three-dimensional problems. For better computational efficiency, operator splitting (Gustafsson et al., 1995) should be used to first decompose the original equation into a series of one-dimensional problems. The new method discussed in this chapter can then be applied to these one-dimensional problems to calculate numerical solutions efficiently. Details will be presented in future papers.

### **7. References**

14 Will-be-set-by-IN-TECH

We apply both the upwind-central scheme and the second-order new method to this example to compare the accuracy of the two algorithms. Table 3 shows the error �*u<sup>e</sup>* <sup>−</sup> *<sup>u</sup>c*�<sup>∞</sup> between the exact solution *u<sup>e</sup>* and the numerical solution *u<sup>c</sup>* calculated using the two algorithms with *κ* = 1.2, for *a* = 0.001 and *a* = 0.0001. All results are obtained using a 3-point stencil. Since the main focus of the study is to compare spatial accuracy of the two algorithms, the first-order explicit time integration is used for simplicity with Δ*t* = 0.0001 to ensure stability. It is clear from Table 3 that the accuracies of the two schemes are as expected: When the grid size Δ*x* is

> Δ*x upwind* 2*nd*-new upwind 2*nd*-new *a* = 0.001 *a* = 0.001 *a* = 0.0001 *a* = 0.0001 1/20 2.69E-04 1.48E-04 3.47E-06 2.31E-06 1/40 1.67E-04 3.87E-05 2.20E-06 6.11E-07 1/80 9.40E-05 9.79E-06 1.23E-06 1.67E-07 1/160 4.98E-05 2.45E-06 6.46E-07 4.13E-08

Table 3. Errors between the exact solution and the numerical solution of Eq. (31) with

The numerical examples presented in this chapter show that the standard central difference scheme is second-order accurate on a 3-point stencil but produces oscillatory solutions for convection dominated problems. The upwind scheme is more robust but is only first-order accurate on a 3-point stencil. The new method discussed in this chapter appears to combine the advantages of accuracy of the standard central difference algorithm with the robustness

The method discussed in this chapter eliminates the convection term in Eq. (1) and makes it feasible to use central difference schemes to solve convection-diffusion equations accurately. The new method, combined with the central difference schemes, can achieve better accuracy than the upwind schemes on the same finite difference stencil, and is shown in the examples presented here to be as robust as the upwind schemes for convection dominated problems. It can also be easily combined with the Padé approximation to achieve fourth-order accuracy in

The new method does incur a modest increase in computational complexity. Instead of just solving Eq. (1), the new method requires solving Eqs. (5) and (7). With a 3-point stencil, the standard upwind-central difference schemes will generate a system of tri-diagonal algebraic equations, while the new method discussed in this chapter will lead to a system of block tri-diagonal algebraic equations with 2 × 2 blocks. This increased complexity, however, can be compensated by the use of fewer grid points with the increased order of accuracy of the new method. Furthermore, for problems that require calculations of both the solutions and their derivatives, the new method eliminates the need to calculate the derivatives after solving

The discussions of the new method in this chapter are based on one-dimensional problems. For higher-dimensional problems, a straightforward application of this method will lead to systems of four equations for two-dimensional problems and seven equations for three-dimensional problems. For better computational efficiency, operator splitting

<sup>2</sup> and ( <sup>1</sup>

<sup>2</sup> )<sup>2</sup> for the upwind-central and

<sup>2</sup> , the errors are reduced by approximately <sup>1</sup>

of the upwind scheme for convection dominated equations.

space on a 3-point finite difference stencil.

reduced by <sup>1</sup>

the 2*nd*−new schemes, respectively.

*a* = 0.001 and *a* = 0.0001.

**6. Conclusions**

Eq. (1).


**5** 

Oleg Druzhinin

*Russia* 

**Internal Waves Radiation by a Turbulent** 

The radiation of internal gravity waves by stratified turbulent shear flows is encountered in many geophysical flows. Numerous applications include jet flows (Sutherland & Peltier 1994), grid-generated turbulence (Dohan & Sutherland 2003), boundary layers (Taylor & Sarkar 2007), collapse of mixed patches (Sutherland et al. 2007), wakes behind towed and self-propelled bodies (Lin & Pao 1979), and many others. The present paper deals with internal waves radiated by a jet flow that is created in the far wake of a sphere towed in a

Experimental studies of internal gravity waves (IW) radiated by a towed sphere and its wake were performed both in a linearly stratified fluid by Bonneton, Chomaz & Hopfinger (1993) (further referred to as BCH) and in the presence of a thermocline by Robey (1996). The results show that internal waves radiated by the sphere (i.e. the lee waves) are stationary with respect to the sphere, and their amplitude is inversely proportional to the sphere Froude number, defined as *Fr V ND* 2 / (where *N* is the buoyancy frequency, and *V* and *D* are the towing speed and the sphere diameter). On the other hand, internal waves, radiated by the turbulent wake are not stationary with respect to the sphere, and their amplitude grows as *Fr* increases. Experimental results obtained by BCH and Robey (1996) show that if the Froude number is sufficiently large (*Fr* > 10), non-stationary IW supercede

Experimental results obtained by BCH show that at early times (*Nt = O*(10)) internal waves are radiated due to the collapse of the vortex coherent structures developing in the near wake (Chomaz, Bonnet & Hopfinger (1993), further referred to as CBH). The wavelength of these waves (also called "random" waves) is of the order of the sphere diameter, and their dynamics is well described theoretically under an assumption that each collapsing coherent structure can be regarded as an impulsive source of IW. Visualization of the density distribution in a horizontal plane at a distance of three sphere radii below the towing axis at times *Nt* < 40 gives a rather complicated, irregular isophase pattern of these random internal

The results of BCH show also that at later times (*Nt* > 40) random internal waves are superceded by waves whose initial spatial period is of the order of five sphere diameters. The isophase distribution of these waves, although being non-stationary with respect to the sphere, is reminiscent of the regular iso-phase pattern of the lee-waves. At sufficiently late times (*Nt* > 50), the random waves disappear, and there remain only coherent IW. BCH give

no explanation of the observed dynamics of these coherent internal waves.

stratified fluid at large Froude and Reynolds numbers.

**1. Introduction** 

the lee waves.

waves.

**Jet Flow in a Stratified Fluid** 

*Institute of Applied Physics, Russian Academy of Sciences* 

