4.2 Characteristics

Characteristics are two sets of intersecting lines in supersonic flow. The characteristics carry a physical significance in that they delineate the region of space that influences flow conditions at a particular point as well as the region of space that depends on the flow conditions at a point. The characteristic lines are selected such that, along these lines, the governing partial differential equations become total differential, finite difference equations, allowing numerical solutions of the flow-field [42, 46].

Alternatively, once a supersonic flow-field has been calculated by some noncharacteristic methods, the characteristic lines can be calculated and superimposed and inferences about influences, causes and effects can be drawn. The α and β or C+ and C� characteristics are inclined at �μ to the local streamlines where μ = sin�<sup>1</sup> (1/M) (Figure 9). In polar coordinates the α and β characteristics'shapes are determined by integrating,

$$\left(\frac{dr}{d\theta}\right)\_{a,\beta} = r \cot \left(\delta - \theta \pm \mu\right) \tag{15}$$

where the plus sign is for the α characteristic and minus is for the β characteristic. For x-y plotting one can integrate the α-characteristics directly:

$$\begin{aligned} (d\mathbf{x}/d\boldsymbol{\theta})\_a &= r\cos\left(\delta + \mu\right) / \cos\left(\pi/2 - \delta - \mu\right) \\ (d\mathbf{y}/d\boldsymbol{\theta})\_a &= r\sin\left(\delta + \mu\right) / \cos\left(\pi/2 - \delta - \mu\right) \end{aligned} \tag{16}$$

Figure 9. Characteristics C<sup>α</sup> and Cβ.

and the β-characteristics by:

$$\begin{aligned} (d\mathbf{x}/d\boldsymbol{\theta})\_\beta &= r\cos\left(\delta-\mu\right)/\cos\left(\pi/2-\delta+\mu\right) \\ (d\mathbf{y}/d\boldsymbol{\theta})\_\beta &= r\sin\left(\delta-\mu\right)/\cos\left(\pi/2-\delta+\mu\right) \end{aligned} \tag{17}$$

zoomed-in schematic of the conditions at (o) showing a Busemann streamline (icfe) that joins a freestream (1) to the conical shock (oe) at (2). A streamline (ss) passes through the shock. The characteristic (os) and its projection to (c) is the last of the centered β-characteristics (blue) and it is also the first of the β-characteristics (red) that start at the surface and proceed towards the axis but intercepts the shock (oe). An examination of the inclinations of the characteristics shows that the angular width of the centered compression fan, ω ¼ μ<sup>2</sup> þ j j δ<sup>2</sup> � μ<sup>1</sup> must be >0, because μ2>μ<sup>1</sup> (since M<sup>2</sup> < M1), so that ω > 0 and the fan must exist. The angular region ω is

characteristics (cfeo). Subscript (1) refers to freestream conditions, (2) refers to pre-shock conditions. Angles

Schematic of characteristics in Busemann flow. Centered compression fan (ioc). Shock-impinging

populated by β characteristics that fan out from (o) to the Busemann streamline along (ic). The fan of β-characteristics contained in (oci) is a centered, axisymmetric compression fan, analogous to the Prandtl-Meyer fan in planar flow. The shape of the last centered characteristic (oc) and the location of (c) can be calculated during the integration of the intake flow when the variable of integration, θ, reaches the value π – (|μ2|+|δ2|). The red β-characteristics from the surface (cfe) all intercept the shock (oe) where a very small, near-apex segment of the shock, is determined by a relatively long length of the Busemann intake surface (ic). The rest of the shock shape is determined by the characteristics from the surface (cfe). This large surface-to-shock length ratio suggests that the leading edge shape is unimportant in determining the overall shock shape. However, a long leading edge surface length contributes to boundary layer growth and viscous losses, providing a reason and an incentive to truncate the leading edge so as to minimize the sum of leading edge shock and boundary layer losses on a practical intake surface. The results presented here give an indication of the extent (ic) to which the conical shock is influenced by a shortening of the intake surface (truncation). A study of viscous/inviscid efficiency loss tradeoffs by truncation or stunting should take direction from the location of point (c). Previous treatment of the centered confocal compression fan or the free-standing conical

An equation for the curvature of the T-M streamline is derived to show that the streamline can have points of zero curvature—inflection points. The Busemann streamline has two points of zero curvature where one of these points has significance in the starting of a Busemann-type intake. The conical surface containing all inflection points in a typical Busemann flow is shown in green in Figure 11 where

<sup>2</sup> An analog of this flow exists in planar flow where the region (ioc) is a Prandtl-Meyer compression fan, the region (cof) is then uniform, the shock (ok) is plane and the flow aft of the shock is again uniform.

shock has not been found in the open literature.2

That is the flow topography in the Prandtl-Meyer intake.

91

Figure 11.

shown are for conditions at O.

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4.4 Surface curvature, D = dδ/ds; inflection point

Integration of the characteristics is easily performed inside the routine for integrating the T-M equations. This method was used to superimpose characteristics on the T-M solution above. Resulting characteristic lines are shown in Figure 10 for the same Mach 5.22 intake as in Figure 6.

### 4.3 Centered compression fan

The Taylor-Maccoll equations point to the existence of a confocal, conical, compression fan—the axisymmetric analogue to a Prandtl-Meyer fan. Such a fan of coalescing characteristics, preceding a free-standing conical shock, is shown to exist experimentally (Figure 8), as well as by CFD calculations (Figures 7a and 10).

The characteristics mesh in Figure 11 is a schematic overlay on the Busemann flow. The α-characteristics (not shown) all start from the freestream Mach cone and proceed away from the axis to intercept either the surface streamline or the front surface of the shock. The blue and red β-characteristics start at the surface and proceed towards the axis. The first of the β-characteristics is the freestream (1) Mach cone itself, having an inclination μ<sup>1</sup> at the axis. At the shock (2) the remaining characteristics have an inclination δ<sup>2</sup> + μ2, different from μ1. Figure 11 is a

#### Figure 10.

α and β characteristics network for the Mach 5.22 Busemann intake. α characteristics are outbound from the center line and β characteristics are inbound. Note convergence of β characteristics at center line.

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#### Figure 11.

and the β-characteristics by:

Hypersonic Vehicles - Past, Present and Future Developments

Figure 9.

Figure 10.

90

Characteristics C<sup>α</sup> and Cβ.

same Mach 5.22 intake as in Figure 6.

4.3 Centered compression fan

ð Þ dx=dθ <sup>β</sup> ¼ r cosð Þ δ � μ = cosð Þ π=2 � δ þ μ ð Þ dy=dθ <sup>β</sup> ¼ rsin ð Þ δ � μ = cosð Þ π=2 � δ þ μ

Integration of the characteristics is easily performed inside the routine for integrating the T-M equations. This method was used to superimpose characteristics on the T-M solution above. Resulting characteristic lines are shown in Figure 10 for the

The Taylor-Maccoll equations point to the existence of a confocal, conical, compression fan—the axisymmetric analogue to a Prandtl-Meyer fan. Such a fan of coalescing characteristics, preceding a free-standing conical shock, is shown to exist experimentally (Figure 8), as well as by CFD calculations (Figures 7a and 10). The characteristics mesh in Figure 11 is a schematic overlay on the Busemann flow. The α-characteristics (not shown) all start from the freestream Mach cone and proceed away from the axis to intercept either the surface streamline or the front surface of the shock. The blue and red β-characteristics start at the surface and proceed towards the axis. The first of the β-characteristics is the freestream (1) Mach cone itself, having an inclination μ<sup>1</sup> at the axis. At the shock (2) the remaining characteristics have an inclination δ<sup>2</sup> + μ2, different from μ1. Figure 11 is a

α and β characteristics network for the Mach 5.22 Busemann intake. α characteristics are outbound from the

center line and β characteristics are inbound. Note convergence of β characteristics at center line.

(17)

Schematic of characteristics in Busemann flow. Centered compression fan (ioc). Shock-impinging characteristics (cfeo). Subscript (1) refers to freestream conditions, (2) refers to pre-shock conditions. Angles shown are for conditions at O.

zoomed-in schematic of the conditions at (o) showing a Busemann streamline (icfe) that joins a freestream (1) to the conical shock (oe) at (2). A streamline (ss) passes through the shock. The characteristic (os) and its projection to (c) is the last of the centered β-characteristics (blue) and it is also the first of the β-characteristics (red) that start at the surface and proceed towards the axis but intercepts the shock (oe).

An examination of the inclinations of the characteristics shows that the angular width of the centered compression fan, ω ¼ μ<sup>2</sup> þ j j δ<sup>2</sup> � μ<sup>1</sup> must be >0, because μ2>μ<sup>1</sup> (since M<sup>2</sup> < M1), so that ω > 0 and the fan must exist. The angular region ω is populated by β characteristics that fan out from (o) to the Busemann streamline along (ic). The fan of β-characteristics contained in (oci) is a centered, axisymmetric compression fan, analogous to the Prandtl-Meyer fan in planar flow. The shape of the last centered characteristic (oc) and the location of (c) can be calculated during the integration of the intake flow when the variable of integration, θ, reaches the value π – (|μ2|+|δ2|). The red β-characteristics from the surface (cfe) all intercept the shock (oe) where a very small, near-apex segment of the shock, is determined by a relatively long length of the Busemann intake surface (ic). The rest of the shock shape is determined by the characteristics from the surface (cfe). This large surface-to-shock length ratio suggests that the leading edge shape is unimportant in determining the overall shock shape. However, a long leading edge surface length contributes to boundary layer growth and viscous losses, providing a reason and an incentive to truncate the leading edge so as to minimize the sum of leading edge shock and boundary layer losses on a practical intake surface. The results presented here give an indication of the extent (ic) to which the conical shock is influenced by a shortening of the intake surface (truncation). A study of viscous/inviscid efficiency loss tradeoffs by truncation or stunting should take direction from the location of point (c). Previous treatment of the centered confocal compression fan or the free-standing conical shock has not been found in the open literature.2

### 4.4 Surface curvature, D = dδ/ds; inflection point

An equation for the curvature of the T-M streamline is derived to show that the streamline can have points of zero curvature—inflection points. The Busemann streamline has two points of zero curvature where one of these points has significance in the starting of a Busemann-type intake. The conical surface containing all inflection points in a typical Busemann flow is shown in green in Figure 11 where

<sup>2</sup> An analog of this flow exists in planar flow where the region (ioc) is a Prandtl-Meyer compression fan, the region (cof) is then uniform, the shock (ok) is plane and the flow aft of the shock is again uniform. That is the flow topography in the Prandtl-Meyer intake.

the portion of the surface (icf) is turning towards the axis and the portion (fk) is turning away.

To derive an expression for the curvature of the T-M streamline we use the defining equation of the streamline,

$$r\,dr/d\theta = r u/v \tag{18}$$

where u and v are the radial and angular components of Mach number as used in the T-M equations. Taking another θ-derivative of (7) gives,

$$\frac{d^2r}{d\theta^2} = -r\frac{u}{v^2}\frac{dv}{d\theta} + \frac{r}{v}\frac{du}{d\theta} + \frac{ru^2}{v^2} \tag{19}$$

In polar coordinates, (r, θ) the curvature of a planar curve is [28, p. 34],

$$D \equiv \left(\frac{\partial \delta}{\partial \mathbf{s}}\right) = \frac{r^2 + 2\left(\frac{dr}{d\theta}\right)^2 - r\frac{d^2r}{d\theta^2}}{\left(r^2 + \left(\frac{dr}{d\theta}\right)^2\right)^{3/2}}\tag{20}$$

3.When u = 0 then D = 0. This means that the streamline has a point of inflection at the place where the radial Mach number is zero. For flow over a cone the condition u = 0 never occurs, so the streamlines are curved monotonically positive. However, for Busemann flow there is a location, θ, where the streamline changes from being concave towards the axis (negative curvature) to being convex (positive curvature). The flow changes from turning inward, towards the axis to turning outward, away from the axis. At the inflected surface there is no turning, the flow is purely convergent. Numerical

Surface curvature (D), pressure gradient (P) and Mach number gradient (dM/ds), vs. axial distance (x) in the isentropic part of the Busemann intake. x = 0 is at the apex of the conical shock. Highest values are reached

Figure 12.

93

at the corner where x = 0.98.

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> integrations of the T-M equations have shown that θ always lies in the interval θ<sup>2</sup> to π/2 (first quadrant), somewhat upstream of the Busemann shock, as shown by the green line in Figure 11. Every Busemann streamline has an inflection point and, for each intake, these points form a unique conical surface. At this angular location the flow is everywhere normal to the green cone surface, whose half-angle is θ, and a conical normal shock can be placed coincident with the green cone since the Mach number is supersonic. This condition leads to an analysis for determining the startability of a wavecatcher Busemann intake according to the following argument: If the bow shock could be coaxed into taking up the inflection position by allowing enough mass spillage to occur between the shock and the inflection location and by restricting the downstream contraction to that allowable by the Kantrowitz criterion for flow starting, then the intake would start. The important variables in the Kantrowitz criterion are the "green" inflection cone surface area, the Mach number in front of the "green" cone and the exit area. These variables are available at the integration of Eqs. (5) and (6) at the streamline inflection angle, θ ¼ θ. If the contraction downstream of the conical normal shock surface does not lead to choking, then the shock moves downstream and the intake starts spontaneously. The starting event and its causes are critical in selfstarting supersonic/hypersonic air intakes. It is a conical and axisymmetric example of the starting criterion posed by Kantrovitz for one-dimensional flow, embodying the same principle of flow choking downstream of a normal shock where, in this case, the normal shock is not flat but has a conical shape. Flow just downstream of the conical normal shock at the inflection point is inclined towards the axis. This (r ! 0)-type singularity is similar to the cone-

Eliminating the derivatives of r with Eqs. (7) and (18) gives,

$$D = \frac{r^2 + 2(ru/v)^2 + r^2 \frac{u}{v^2} \frac{dv}{d\theta} - \frac{r^2}{v} \frac{du}{d\theta} - (ru/v)^2}{(r^2 + r^2 u^2/v^2)^{3/2}} \tag{21}$$

In this expression the derivatives dv/dθ and du/dθ are given by the Taylor-Maccoll Eqs. (5) and (6) so that the streamline curvature is,

$$D \equiv \frac{d\delta}{ds} = \frac{uw(u + v \cot \theta)}{r(v^2 - 1)(v^2 + u^2)^{3/2}} \tag{22}$$

This equation gives the curvature of the T-M streamline in terms of the polar coordinates, r and θ, and the radial and polar Mach number components, u and v. It is plotted as the black curve in Figure 12. A number of very interesting and important features, about the T-M streamline, become apparent from an examination of its curvature as given by Eq. (21):


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#### Figure 12.

the portion of the surface (icf) is turning towards the axis and the portion (fk) is

To derive an expression for the curvature of the T-M streamline we use the

where u and v are the radial and angular components of Mach number as used in

u v2 dv dθ þ r v du dθ þ

In polar coordinates, (r, θ) the curvature of a planar curve is [28, p. 34],

<sup>¼</sup> <sup>r</sup><sup>2</sup> <sup>þ</sup> <sup>2</sup> dr dθ <sup>2</sup> � <sup>r</sup> <sup>d</sup><sup>2</sup>

<sup>2</sup> <sup>þ</sup> <sup>r</sup><sup>2</sup> <sup>u</sup> v2 dv <sup>d</sup><sup>θ</sup> � <sup>r</sup><sup>2</sup> v du <sup>d</sup><sup>θ</sup> � ð Þ ru=v

ds <sup>¼</sup> uv uð Þ <sup>þ</sup> <sup>v</sup> cot <sup>θ</sup>

This equation gives the curvature of the T-M streamline in terms of the polar coordinates, r and θ, and the radial and polar Mach number components, u and v. It is plotted as the black curve in Figure 12. A number of very interesting and important features, about the T-M streamline, become apparent from an examination of

1. D is inversely proportional to r so that when r ! 0 then D ! ∞. This means that streamlines near the origin of T-M flows are highly curved. This is a necessary condition for flow over a cone, where flow, near the tip and just aft of the conical shock, has to rapidly adjust to the inclination demanded by the cone surface, since the flow deflection produced by the conical shock is insufficient for the flow to be tangent to the cone surface. Similar highly curved streamlines are to be expected near the focal point of Busemann flow. Conical flow is not conically symmetric, i.e., independent of r, when it comes to gradients of its dependent variables, such as streamline curvature—the dependence being inversely proportional to r. This inverse dependence on r

2. There is an asymptotic condition, (D = 0) in the T-M streamlines at v = 0. For flow over a cone, v = 0 at the cone surface. This confirms that the streamlines become asymptotic to the cone surface as they approach the surface. There is

<sup>r</sup><sup>2</sup> <sup>þ</sup> dr dθ

the T-M equations. Taking another θ-derivative of (7) gives,

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d2 r <sup>d</sup>θ<sup>2</sup> ¼ �<sup>r</sup>

<sup>D</sup> � <sup>∂</sup><sup>δ</sup> ∂s 

<sup>D</sup> <sup>¼</sup> <sup>r</sup><sup>2</sup> <sup>þ</sup> <sup>2</sup>ð Þ ru=<sup>v</sup>

<sup>D</sup> � <sup>d</sup><sup>δ</sup>

extends to other flow property gradients as well.

no v = 0 asymptotic condition in Busemann flow.

its curvature as given by Eq. (21):

92

Eliminating the derivatives of r with Eqs. (7) and (18) gives,

In this expression the derivatives dv/dθ and du/dθ are given by the Taylor-Maccoll Eqs. (5) and (6) so that the streamline curvature is,

dr=dθ ¼ ru=v (18)

<sup>v</sup><sup>2</sup> (19)

ru<sup>2</sup>

r dθ<sup>2</sup>

<sup>2</sup> <sup>3</sup>=<sup>2</sup> (20)

2

<sup>r</sup><sup>2</sup> <sup>þ</sup> <sup>r</sup><sup>2</sup>u<sup>2</sup>=v<sup>2</sup> ð Þ<sup>3</sup>=<sup>2</sup> (21)

r vð Þ <sup>2</sup> � <sup>1</sup> <sup>v</sup><sup>2</sup> <sup>þ</sup> <sup>u</sup><sup>2</sup> ð Þ<sup>3</sup>=<sup>2</sup> (22)

turning away.

defining equation of the streamline,

Surface curvature (D), pressure gradient (P) and Mach number gradient (dM/ds), vs. axial distance (x) in the isentropic part of the Busemann intake. x = 0 is at the apex of the conical shock. Highest values are reached at the corner where x = 0.98.

3.When u = 0 then D = 0. This means that the streamline has a point of inflection at the place where the radial Mach number is zero. For flow over a cone the condition u = 0 never occurs, so the streamlines are curved monotonically positive. However, for Busemann flow there is a location, θ, where the streamline changes from being concave towards the axis (negative curvature) to being convex (positive curvature). The flow changes from turning inward, towards the axis to turning outward, away from the axis. At the inflected surface there is no turning, the flow is purely convergent. Numerical integrations of the T-M equations have shown that θ always lies in the interval θ<sup>2</sup> to π/2 (first quadrant), somewhat upstream of the Busemann shock, as shown by the green line in Figure 11. Every Busemann streamline has an inflection point and, for each intake, these points form a unique conical surface. At this angular location the flow is everywhere normal to the green cone surface, whose half-angle is θ, and a conical normal shock can be placed coincident with the green cone since the Mach number is supersonic. This condition leads to an analysis for determining the startability of a wavecatcher Busemann intake according to the following argument: If the bow shock could be coaxed into taking up the inflection position by allowing enough mass spillage to occur between the shock and the inflection location and by restricting the downstream contraction to that allowable by the Kantrowitz criterion for flow starting, then the intake would start. The important variables in the Kantrowitz criterion are the "green" inflection cone surface area, the Mach number in front of the "green" cone and the exit area. These variables are available at the integration of Eqs. (5) and (6) at the streamline inflection angle, θ ¼ θ. If the contraction downstream of the conical normal shock surface does not lead to choking, then the shock moves downstream and the intake starts spontaneously. The starting event and its causes are critical in selfstarting supersonic/hypersonic air intakes. It is a conical and axisymmetric example of the starting criterion posed by Kantrovitz for one-dimensional flow, embodying the same principle of flow choking downstream of a normal shock where, in this case, the normal shock is not flat but has a conical shape. Flow just downstream of the conical normal shock at the inflection point is inclined towards the axis. This (r ! 0)-type singularity is similar to the conetip singularity described above; its existence, in the idealized form, has not seen confirmation by experiment or CFD. The cone of streamline inflections is a significant feature for assessing startability of wavecatcher Busemann intakes in Section 8.

dM

u du

v dv

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ds <sup>¼</sup> <sup>v</sup>

dM

number and pressure are related by [47],

so that,

where dM

95

ds <sup>¼</sup> <sup>v</sup>

Maccoll Eqs. (5) and (6), when multiplied by u and v, respectively.

<sup>d</sup><sup>θ</sup> <sup>¼</sup> uv <sup>þ</sup> <sup>γ</sup> � <sup>1</sup>

<sup>d</sup><sup>θ</sup> ¼ �uv <sup>þ</sup> <sup>v</sup> <sup>1</sup> <sup>þ</sup> <sup>γ</sup> � <sup>1</sup>

r u<sup>2</sup> <sup>þ</sup> <sup>v</sup><sup>2</sup> ð Þ <sup>1</sup> <sup>þ</sup> <sup>γ</sup> � <sup>1</sup>

4.6 Surface pressure gradient in the flow direction, P = (dp/ds)/(ρV<sup>2</sup>

dM

dM

or

<sup>P</sup> � dp=ds

r u<sup>2</sup> <sup>þ</sup> <sup>v</sup><sup>2</sup> ð Þ <sup>u</sup>

where the derivative terms, in the square brackets, are given by the Taylor-

<sup>2</sup> <sup>u</sup><sup>2</sup> v

This is the Mach number gradient, expressed in terms of the coordinates (r, θ) and the corresponding Mach number components (u, v), where the Mach number component values come directly from the integration of the Taylor-Maccoll Eqs. (5) and (6). dM/ds is plotted in Figure 12 (with s measured in the downstream direction).

In the isentropic flow, from the freestream to the shock, the gradients of Mach

<sup>2</sup> <sup>M</sup><sup>2</sup> γM<sup>2</sup>

2

ds is given by Eq. 26; so that the non-dimensional pressure gradient is,

dp p

M<sup>2</sup> MP

ð Þ u þ v cot θ

<sup>v</sup>ð Þ <sup>2</sup> � <sup>1</sup> (30)

<sup>M</sup> ¼ � <sup>1</sup> <sup>þ</sup> ð Þ <sup>γ</sup>�<sup>1</sup>

<sup>ρ</sup>V<sup>2</sup> <sup>¼</sup> �v<sup>2</sup>

<sup>3</sup> These gradient equations are applicable to all types of Taylor-Maccoll flows [48].

ds ¼ � <sup>1</sup> <sup>þ</sup> ð Þ <sup>γ</sup> � <sup>1</sup>

r u<sup>2</sup> <sup>þ</sup> <sup>v</sup><sup>2</sup> ð Þ<sup>2</sup>

The pressure gradient is expressed in terms of the radial and azimuthal coordinates r and θ and the radial and angular Mach number components u and v. It is plotted in Figure 12. This permits the calculation of the surface pressure gradient from quantities obtained in the T-M calculation of Busemann flow.3 It also means that the surface pressure gradients are known everywhere on the surface of the highly three-dimensional wavecatcher shapes where all the surface gradients are useful as inputs to boundary layer calculations. Towards this end it is noted that, for the flow just upstream of the corner, where the shock impinges, the u and v Mach numbers are given by u<sup>2</sup> and v<sup>2</sup> from Eqs. (11) and (12), so that the gradients immediately before the shock-boundary-layer interaction at the corner can be evaluated just from the prescribed initial conditions using Eqs. (5)–(7), before embarking on a calculation of Eqs. (5) and (6). This enables a selection of initial conditions that is based on considerations involving the shock losses as well as the shock-boundary-layer interaction

<sup>2</sup> <sup>v</sup><sup>2</sup> <sup>u</sup> <sup>þ</sup> <sup>v</sup> cot <sup>θ</sup>

<sup>2</sup> <sup>u</sup><sup>2</sup> <sup>þ</sup> <sup>v</sup><sup>2</sup> <sup>u</sup> <sup>þ</sup> <sup>v</sup> cot <sup>θ</sup>

du <sup>d</sup><sup>θ</sup> <sup>þ</sup> <sup>v</sup> dv dθ

u þ v cot θ

(26)

<sup>v</sup><sup>2</sup> � <sup>1</sup> (27)

<sup>v</sup><sup>2</sup> � <sup>1</sup> (28)

<sup>v</sup><sup>2</sup> � <sup>1</sup> (29)

)

