Effects of the suction/injection and external free stream on the instability of a boundary layer over a rotating disk

Al Saeedi, Bashar; Abdelrazik, Fathy; Fildes, Matthew; Hussain, Zahir

doi:10.1063/5.0137448

In this paper, we investigated the effectiveness of the strength of axial flow and suction/injection for the viscous mode (type II) instability within the boundary layer of a rotating disk. To investigate the lower branch, we scaled the viscous mode by using a familiar triple-deck structure analogous to that which was found for Blasius flow over a flat plate. We also analyzed the linear stability behavior of high-Reynolds laminar-turbulent transition. To describe the stationary type II wavenumber and waveangle, we conducted an asymptotic analysis followed by a comparison with the type I mode. We found that a positive axial flow had a stabilizing effect and vice versa a negative axial flow exhibited a destabilizing effect. The results were consistent with previous studies in the literature for positive axial flow, as disturbances were advected downstream in the radial direction. Regarding the suction/injection study, we found suction to be stabilizing, which aligns with previous results in the literature although, conversely, the injection was found to be destabilizing. For the numerical analysis, we found that parameters which result in an increase or decrease of the critical Reynolds number led to a stabilization or destabilization of the flow, respectively. Finally, we compared the asymptotic and numerical stability results for both types I and II followed by the critical Reynolds numbers comparisons, which were found to be consistent in general with results in the literature.

I. INTRODUCTION

For many decades, the subject of the rotating disk within the boundary layer has been of great interest; in particular, the fundamental role of the disk as a model for cross-flow dominated flows over swept wings and other applications. Even though disk flow has been studied extensively since the 1950s, interest in boundary layer flows over axisymmetric rotating bodies has increased in the past couple of decades. For example, the advances in engineering in aero-engine components and spinning projectiles have led to the study of fluid flow over rotating cones and hemispheres being of direct industrial importance. It could, therefore, be possible to improve engineering designs in a number of different industrial sectors by improving our understanding the transition of the laminar-turbulent over these geometries as an example. Many studies focused on the manner mechanisms that govern the laminar-turbulent transition and also analyzed the instability of flow in three-dimensional (3D) boundary layers in detail.^1–7 In addition, early studies investigated the incompressible flow over rotating disks^8,9 to most recent studies.^10–14 Other experimental studies investigated the incompressible flow over rotating cones.^15–18 All these studies led to the recent theoretical studies.^19–22

Schlichting²³ and Tollmien²⁴ studied the effects of linear and disturbance-induced waves on laminar flow over flat plates. They considered the amplification of such disturbances as the primary cause of a flow instability. Gregory et al.⁸ then investigated the formation of co-rotating stationary vortices (crossflow instabilities) on a rotating disk. From a physical perspective in a stationary fluid, a rotating disk flow (for fluid surrounding the rotating disk) is considered a centrifugal pump. The viscous effects cause this fluid to rotate. After flowing radially outward, the fluid is castoff at the edge of the disk after being entrained toward the surface. By finding exact similar solutions to the Navier–Stokes equations for the infinite plane rotating disk problem, Von Kármán²⁵ resulted in a new set of ordinary differential equations that described the mean flow for the disk. His study was built on understanding the concept of instability mechanisms Coriolis forces. These forces are operational to a breakdown of the boundary layer on a swept wing rotating disk. By using a numerical analysis in order to derive the neutral curves for the stationary instability modes in the rotating disk boundary layer, Malik¹⁰ had uncovered an additional instability mechanism. This was the first time that a sixth-order system with rotational Coriolis and viscous terms was used instead of the earlier fourth-order Orr–Sommerfeld type studies. In addition, Hall³ verified asymptotically that two branches might exist: an upper branch which is related to the crossflow instability (type I) and a lower branch which is related to the streamline curvature and balancing the Coriolis and viscous forces (type II). The inviscid and viscous modes are governed by the crossflow instability. The former is known to arise at an inflection point in the radial mean velocity profile. Hence, at the location of the inflection point, vortex activity is located in a critical layer, and the perturbation wavelengths are determined by the thickness of the boundary layer.⁸ While the latter corresponds to the effective velocity profile that exhibits zero shears at the wall.³

Previous studies showed that an appropriate Reynolds number could indicate the stability strength of the flow at a point in the transition from the laminar to the turbulent regime. Recent works on the linear stability of flows have been influenced by early emerging studies. Because Brown²⁶ disagreed that some experimental studies ignored the curvature effects, they started to be considered in successive studies; for example, Malik et al.²⁷ successfully determined the Reynolds number to be 287 demonstrating a strong correlation with the experimental results obtained by Wilkinson and Malik.²⁸

Turkyilmazoglu and Gajjar²⁹ analyzed the rotating disk's traveling mode—which relied on the numerical and asymptotic study of the boundary layer on a rotating disk with stationary modes conducted in the previous reports^5,10—by analyzing linear absolute/convective instability of flow due to a rotating disk. It also confirmed previous studies by Balakumar and Malik³⁰ who examined traveling disturbances on the rotating disk flow. Balakumar and Malik's traveling mode perturbation equations were solved by a spectral approach, which is computed numerically and asymptotically that allowed the wavenumber quantities to be solved separately. However, both of them solved the eigenvalues and Reynolds numbers values to scale the frequencies of traveling modes. Hence, these values helped them compare different purposes of travel modes. Turkyilmazoglu³¹ also solved type II viscous modes by using the non-linear method of a rotating disk resulting in the frequency of the positive value to show the branch II. He also described this as a representation of Fedorov et al.³² findings in his experimental study of type III eddies. Although the type I mode asymptotic derivation is not covered, it analyzed the type II non-linear modes. Hussain et al.³³ used the similar scaling methods that Hussain⁷ presented in his thesis to cover inviscid and viscous modes. They observed that the traveling modes had large and lighter effects on types I and II, respectively. Similarly, using the same frequency as Turkyilmazoglu and Gajjar²⁹ used, they compared asymptotic output results to numerical analysis for both types I and II and the reason behind this is due to the various scalings used for each case. Meanwhile, they observed that the asymptotic methods for type II had frequency values up to 4. Garrett and Peake¹⁹ expanded the boundary layer theoretically by studying the absolute stability for a rotating cone, where they used local stability to find the critical Reynolds number values for the absolute instability.

Recently, Thomas and Davis³⁴ updated the analysis of the absolute instability. They calculated this absolute instability by using the global stability analysis of the case of a rotating cone. According to their results, large azimuthal wave numbers showed faster exponential growth and required a greater azimuthal wave number than the critical Reynolds number for absolute instability. However, by using Garrett and Peake¹⁹ results, Garret et al.²⁰ predicted the wavenumber and waveangle** of instability waves on boundary layers for the case of a rotating cone, by using asymptotic and numerical analysis. In this paper, the asymptotic analysis used follows presented by Hall.³ They observed that the inviscid mode stabilized the flow when the half angle increased and balanced the viscous mode due to Coriolis force. Hussain's⁷ study covered the asymptotic analysis for a rotating cone in two cases (still fluid and axial flow) for both types I and II. Garret et al.²⁰ used this to cover both traveling mode and stationary growth rates resulting in type II stationary modes becoming less amplified when increasing half angle. In addition, they found the traveling modes had a larger effect on the type II modes and had very little effect on the type I modes for the rotating disk. This result is consistent with previous findings in the rotating disk. However, the rotating disk is considered a special case of the rotating cone set to 90°. Recently, Fildes et al.³⁵ compared the asymptotic vs numerical analysis effective traveling modes wavenumber and waveangle for both types I and II for the case of a rotating cone in still fluid. Hussain⁷ investigated the type II mode in the case of increasing the strength axial flow $T_{s} = 0 - 0.25$ ⁠. In this study, we continued our previous work on the rotating disk³³ via investigating the stability of stationary viscous modes by first adding positive/negative strength of axial flow T_s and, second, by adding various suction/injection parameters $W^{*}$ with the fixed axial flow at $T_{s} = 0.5$ ⁠. We also used the same methods that Fildes et al. used to compare the asymptotic and numerical analysis for a rotating cone.

Overall, in Secs. II and III, we formulated the linear disturbance equations to identify the lower branch viscous modes linear stability followed by the asymptotic results. Section IV shows the asymptotic analysis results for type II and compares them to the findings by Al Saeedi and Hussain¹⁴ obtained for type I mode. In Sec. V, we computed neutral stability curves for both type I and II modes. Section VI concludes with a comparison between asymptotic and numerical analysis for both type I and II modes. Finally, in Sec. VII, we discuss the location of the critical Reynolds number of both suction/injection within various axial flows.

II. PROBLEM FORMULATION

In this paper, we followed the formulations of Al Saeedi and Hussain¹⁴ for the rotating disk to investigate the asymptotic and numerical analysis. We also summarized the formulation used in the asymptotic analysis and noted the amendments to the formulation of the numerical analysis where relevant. Imagine a smooth rigid disk which is rotating about the

z^{*}

axis. The coordinates radial r and azimuthal θ rotate with the disk surface (Fig. 1). In an incompressible fluid, the disk is placed with the oncoming strength of the axial flow which is parallel to

z^{*}

axis at upstream infinity. Along the disk, the dimensional surface velocity distribution at the boundary-layer edge is given by

U_{0}^{*} = C^{*} r^{*}

the potential-flow solution (see, for example, Refs. 36 and 37). On the disk, the scale factor

C^{*}

is determined by the free-stream strength of axial flow. The formulation diagram is shown in Fig. 1 of Al Saeedi and Hussain.¹⁴ Following the details of the asymptotic non-dimensionalization in Ref. 14 leads to the expression for the Reynolds number,

R = \frac{Ω^{*} l^{* 2}}{v^{*}} .

(1)

FIG. 1.

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A diagram representing a rotating disk model setup in axial flow with suction and injection for coordinates in the radial r and azimuthal θ directions.

Ω^{*}

describes the angular speed of the disk rotation,

l^{*}

is a characteristic length scale which is along the disk surface, and

v^{*}

describes the kinematic viscosity of the fluid. In the

z^{*}

direction, the distances are scaled

δ^{*} = {(v^{*} / Ω^{*})}^{1 / 2}

on the boundary layer thickness; this leads to a non-dimensional variable

η = z^{*} / δ^{*}

⁠. We note that

O (R^{- 1 / 2})

is a boundary layer thickness. For the basic steady flow, the velocity scales in the radial, azimuthal, and normal directions are given by

u = u_{b} = Ω^{*} l^{*} (rU (η; T_{s}), rV (η; T_{s}), R^{- 1 / 2} W (η; T_{s})) .

(2)

While the pressure of the basic flow is

p^{*} = p_{b} = ρ^{*} Ω^{*} l^{* 2} (P_{0} (r) + R^{- 1 / 2} r P (η; T_{s})),

(3)

with

P_{0} (r) = - \frac{C^{* 2} r^{2}}{2 Ω^{* 2}},

(4)

the inviscid Bernoulli pressure between inside the boundary layer and outside coming from the free-stream flow. Then, these velocity scales are determined by the continuity and Navier–Stokes equations,

W' + 2 U = 0,

(5)

W U' + U^{2} - {(V + 1)}^{2} = T_{s}^{2} + U^{″},

(6)

W V' + 2 U (V + 1) = V^{″} .

(7)

With boundary conditions,

U = 0, V = 0, W = W^{*}, on η = 0,

(8)

{U \to T_{s}}, V \to - 1, as η \to \infty .

(9)

With respect to η, the prime denotes differentiation. The ratio of the local slip velocity is the parameter T_s considered at a radial position in relation to the rotational speed of the disk surface

T_{s} = \frac{C^{*}}{Ω^{*}}

⁠. Here, T_s is independent of

r^{*}

simplifying the analysis for the rotating disk. By using integration methods of a fourth-order Runge–Kutta scheme, the above Eqs. are solved. However, Al Saeedi and Hussain¹⁴ changed the location of the edge of the boundary layer which confirmed the convergent solutions. Their results showed the radial, azimuthal, and normal to the wall components of the steady mean flow velocity of the disk in the boundary layer for the two cases: positive/negative strength of axial flow T_s and various suction/injection parameters

W^{*}

within the fixed axial flow.¹⁴

III. ASYMPTOTIC LINEAR DISTURBANCE EQUATIONS

To investigate the linear stability analysis for type II modes, we derive the wavenumber

γ_{δ^{*}}

and waveangle

ϕ

estimates of resulting disturbances in the limit of large Reynolds numbers. This is analogous to the investigation that Al Saeedi and Hussain¹⁴ achieved for type I modes. Following them, we must obtain the linear disturbance equations by introducing small perturbations on both the mean flow profile and fluid pressure; these perturbations have to be around the basic flow that leads to equations independent of T_s which are similar to equations are given in Ref. 14 as

u = u_{b} + \tilde{u}, p^{*} = p_{b}^{*} + \tilde{p^{*}},

(10)

where

\tilde{u} = Ω^{*} (\tilde{u}, \tilde{v}, \tilde{w}), \tilde{p^{*}} = (ρ^{*} Ω^{* 2} l^{*}) \tilde{p} .

(11)

From Eqs. ,¹⁴ we must non-dimensionalize and remove non-linear terms,

\frac{\partial \tilde{u}}{\partial r} + \frac{\tilde{u}}{r} + \frac{1}{r} \frac{\partial \tilde{v}}{\partial θ} + \frac{\partial \tilde{w}}{\partial z} = 0,

(12)

\begin{matrix} (r U \frac{\partial}{\partial r} + V \frac{\partial}{\partial θ} + R^{- \frac{1}{2}} W \frac{\partial}{\partial z}) \tilde{u} + U \tilde{u} + r \tilde{w} \frac{\partial U}{\partial z} - 2 (V + 1) \tilde{v} \\ = - \frac{\partial \tilde{p}}{\partial r} + \frac{1}{R} (\nabla^{2} \tilde{u} - \frac{\tilde{u}}{r^{2}} - \frac{2}{r^{2}} \frac{\partial \tilde{v}}{\partial θ}), \end{matrix}

(13)

\begin{matrix} (r U \frac{\partial}{\partial r} + V \frac{\partial}{\partial θ} + R^{- \frac{1}{2}} W \frac{\partial}{\partial z}) \tilde{v} + V \tilde{u} + r \tilde{w} \frac{\partial V}{\partial z} + 2 (V + 1) \tilde{u} \\ = - \frac{1}{r} \frac{\partial \tilde{p}}{\partial θ} + \frac{1}{R} (\nabla^{2} \tilde{v} + \frac{2}{r^{2}} \frac{\partial \tilde{u}}{\partial θ} - \frac{\tilde{v}}{r^{2}}), \end{matrix}

(14)

\begin{matrix} (r U \frac{\partial}{\partial r} + V \frac{\partial}{\partial θ} + R^{- \frac{1}{2}} W \frac{\partial}{\partial z}) \tilde{w} + R^{- \frac{1}{2}} \tilde{w} \frac{\partial W}{\partial z} \\ = - \frac{\partial \tilde{p}}{\partial z} + \frac{1}{R} (\nabla^{2} \tilde{w}) . \end{matrix}

(15)

The quantities

\nabla^{2} = l^{*} \nabla^{* 2}

and

r = \frac{r^{*}}{l^{*}}

are the non-dimensional Laplacian for the above perturbation equations. Following Hussain,⁷ who investigated the type II mode when increasing the strength axial flow,

T_{s} = 0 - 0.25

⁠. We consider the stability of stationary viscous modes by first adding the positive/negative axial flow strength T_s. Second adding various suction/injection parameters

W^{*}

with a fixed axial flow that Hussain obtained at

T_{s} = 0.5

⁠. By applying a triple-deck structure flow, originally established by Blasius, over a flat plate,³⁸ we scaled the lower branch disturbance. Moreover, regarding the time-independent stationary modes investigation condition, the effective wall shear (⁠

α U' r + β V'

⁠) at the leading order is considered to be zero. In contrast to the earlier conditions of inviscid stability analysis, to remove this condition at the critical layer location

η = \bar{η}

⁠, we required the effective velocity

\bar{\bar{U}}

and its second derivative,

{\bar{\bar{U}}}^{″}

⁠. Therefore, we proceeded to construct our structure of triple-deck by using the convenient small parameters which is given by

ε = R^{- \frac{1}{16}} .

(16)

Meanwhile, the thickness order of lower, main and upper decks is typically

ε^{9}, ε^{8}

⁠, and

ε^{4}

⁠, respectively. According to Hall,³ the triple-deck arrangement identifying the viscous wall modes on a rotating disk is illustrated in Fig. 2. In order to represent the lower, main and upper O(1) variation, we defined ξ, ζ, and Z, respectively, as inner variables. Moreover, by using the setup of Hall, the radial and azimuthal wavenumbers α and β are scaled regarding a viscous length scale. Consequently, the perturbation of velocity becomes

\tilde{u} = u (z) exp (\frac{i}{ε^{4}} {\int^{r} α (r, ε) d r + β (ε) θ}),

(17)

with the similar expressions for the pressure perturbation

\tilde{p}

as well as

\tilde{w}

and

\tilde{v}

⁠. We also expanded the radial and azimuthal wavenumbers as

α = α_{0} + ε^{2} α_{1} + ε^{3} α_{2} + \dots,

(18)

β = β_{0} + ε^{2} β_{1} + ε^{3} β_{2} + \dots .

(19)

FIG. 2.

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Schematic diagram shows the triple-deck structure.

Taking into consideration that the order ε terms here are zero. The necessity of stationary modes is to obtain $β_{i}, α_{i} \in R$ ⁠, so at the location r, the flow is neutrally stable. Interestingly, the wall-dominated type II mode analysis in this study for the rotating disk has shown up to be very similar to the Hall³ and Hussain⁷ studies regarding rotating disk wall modes. Following the analysis of Hall and Hussain, we modified some terms in the upper deck disturbance equations pertaining to the introduction of an axial flow and applying suction/injection in a similar way to the previously described type I modes investigation.

A. Leading order triple-deck solution

For the positive/negative strength of axial flow T_s and various suction/injection parameters

W^{*}

within the fixed axial flow, the basic flow on the upper deck expands as U = T_s, V = −1. Meanwhile, the disturbance fields have expansions of the form

u = ε^{3} u_{0}^{U} (Z) + ε^{4} u_{1}^{U} (Z) + \dots,

(20)

v = ε^{3} v_{0}^{U} (Z) + ε^{4} v_{1}^{U} (Z) + \dots,

(21)

w = ε^{3} w_{0}^{U} (Z) + ε^{4} w_{1}^{U} (Z) + \dots,

(22)

p = ε^{3} p_{0}^{U} (Z) + ε^{4} p_{1}^{U} (Z) + \dots .

(23)

The obtained equations by balancing terms of

O (ε^{- 1})

in Eqs. are given by

i (α_{0} u_{0}^{U} + \frac{β_{0} v_{0}^{U}}{r}) + \frac{d w_{0}^{U}}{d Z} = 0,

(24)

(β_{0} - α_{0} r T_{s}) u_{0}^{U} = α_{0} p_{0}^{U},

(25)

(β_{0} - α_{0} r T_{s}) v_{0}^{U} = \frac{β_{0} p_{0}^{U}}{r},

(26)

i (β_{0} - α_{0} r T_{s}) w_{0}^{U} = \frac{d p_{0}^{U}}{d Z} .

(27)

Substituting for

u_{0}^{U}

and

v_{0}^{U}

from Eqs. and into , then using Eq. results in

\frac{d^{2} w_{0}^{U}}{d Z^{2}} - γ_{0}^{2} w_{0}^{U} = 0,

(28)

where

γ_{0} = \sqrt{(α_{0}^{2} + \frac{β_{0}^{2}}{r^{2}})}

is the leading order wavenumber excluding the following case:

(β_{0} - α_{0} r T_{s}) = 0.

(29)

Analyzing this system of equations as

z \to \infty

yields similar solutions to Hall,³ indicated by

u_{0}^{U} = \frac{α_{0} C}{(β_{0} - α_{0} r T_{s})} e^{- γ_{0} Z},

(30)

v_{0}^{U} = \frac{α_{0} C}{r (β_{0} - α_{0} r T_{s})} e^{- γ_{0} Z},

(31)

w_{0}^{U} = \frac{i γ_{0} C}{(β_{0} - α_{0} r T_{s})} e^{- γ_{0} Z},

(32)

p_{0}^{U} = C e^{- γ_{0} Z},

(33)

where

C =

const. We scale the radial and azimuthal disturbance velocity expansions in the main deck. This is achieved by the difference in powers of ε between the upper and main deck. In order to match with the upper deck, the pressure disturbance expansions and surface normal at the same leading order are both kept giving

u = ε^{- 1} u_{0}^{M} (ζ) + u_{1}^{M} (ζ) + \dots,

(34)

v = ε^{- 1} v_{0}^{M} (ζ) + v_{1}^{M} (ζ) + \dots,

(35)

w = ε^{- 1} w_{0}^{M} (ζ) + w_{1}^{M} (ζ) + \dots,

(36)

p = ε^{- 1} p_{0}^{M} (ζ) + p_{1}^{M} (ζ) + \dots .

(37)

Noting that

Z \to 0, p_{0}^{U} \to C

as well as by Prandtl matching that

lim_{ζ \to \infty} p_{0}^{M} (ζ) = lim_{Z \to 0} p_{0}^{U} (Z) = C .

(38)

By equating terms of

O (ε^{- 5})

and expanding the disturbance equations lead to

i α_{0} u_{0}^{M} + \frac{i β_{0} v_{0}^{M}}{r} + \frac{d w_{0}^{M}}{d ζ} = 0,

(39)

i (α_{0} r U + β_{0} V) u_{0}^{M} + r U' w_{0}^{M} = 0,

(40)

i (α_{0} r U + β_{0} V) v_{0}^{M} + r V' w_{0}^{M} = 0,

(41)

\frac{d p_{0}^{M}}{d ζ} = 0.

(42)

We can see that

p_{0}^{M} (ζ) =

const from Eq. . Therefore, using Eq. gives

p_{0}^{M} = C

⁠. Consequently, eliminating

v_{0}^{M}

and

u_{0}^{M}

from Eq. , preferably before integrating the resulting expression with regard to ζ, results in the disturbance field of the main deck. It is similar to Hall³ findings and also it matches with the previous solutions of the upper deck.

u_{0}^{M} = \frac{C r γ_{0} U'}{{(β_{0} - α_{0} r T_{s})}^{2}},

(43)

v_{0}^{M} = \frac{C r γ_{0} V'}{{(β_{0} - α_{0} r T_{s})}^{2}},

(44)

w_{0}^{M} = \frac{iCr γ_{0}}{{(β_{0} - α_{0} r T_{s})}^{2}} (α_{0} r U + β V_{0}) .

(45)

Here,

w_{0}^{M}

fulfills the no-slip condition as

ζ \to 0

⁠, with the effective wall shear decaying to zero, that is

α_{0} r U' + β_{0} V' \to 0,

(46)

as

ζ \to 0

⁠. Now, if we choose the leading order radial and azimuthal wavenumbers such that

α_{0} U' (0) + \frac{β_{0}}{r} V' (0) = 0,

(47)

then we have

α_{0} u_{0}^{M} + \frac{β_{0}}{r} v_{0}^{M} \to 0

as

ζ \to 0

⁠. The constraint Eq. forces us to consider only neutral, or stationary, disturbances. In our rotating coordinate system, these modes stay fixed on the surface. For the positive/negative strength of axial flow T_s and various suction/injection parameters

W^{*}

within a fixed axial flow, we know

U'

and

V'

⁠, from Malik²⁷ giving

\frac{β_{0}}{α_{0} r}

⁠. Importantly, when

T_{s} = T_{s c} = | \frac{U' (0)}{V' (0)} |,

(48)

the analysis breaks down. However, we notice from Tables I–III that regarding the axial flow parameters range, we investigated the positive/negative strength of the axial flow T_s and suction/injection

W^{*}

with the fixed axial flow

T_{s} = 0.05

⁠.

TABLE I.

Investigating whether $T_{s} c = | U' (0) / V' (0) |$ for a rotating disk in axial flow $T_{s} > 0.25$ ⁠.

*T_s*	$U' (0)$	$V' (0)$	$\| T_{s c} \|$
0.3	0.6219	−0.7359	0.84508
0.35	0.6615	−0.7628	0.8671
0.4	0.7066	−0.7904	0.8939
0.45	0.7567	−0.8183	0.9247
0.5	0.8116	−0.8462	0.9591
0.55	0.871	−0.8741	0.9966
0.6	0.9347	−0.9017	1.0365
0.65	1.0024	−0.9291	1.0788

*T_s*	$U' (0)$	$V' (0)$	$\| T_{s c} \|$
0.3	0.6219	−0.7359	0.84508
0.35	0.6615	−0.7628	0.8671
0.4	0.7066	−0.7904	0.8939
0.45	0.7567	−0.8183	0.9247
0.5	0.8116	−0.8462	0.9591
0.55	0.871	−0.8741	0.9966
0.6	0.9347	−0.9017	1.0365
0.65	1.0024	−0.9291	1.0788

TABLE II.

Investigating whether $T_{s} c = | U' (0) / V' (0) |$ for a rotating disk in axial flow $T_{s} < 0$ ⁠.

T_s	$U' (0)$	$V' (0)$	$\| T_{s c} \|$
−0.01	0.5104	−0.6145	0.8305
−0.02	0.5018	−0.6138	0.8175
−0.03	0.5115	−0.6131	0.8342
−0.04	0.5121	−0.612	0.8366

T_s	$U' (0)$	$V' (0)$	$\| T_{s c} \|$
−0.01	0.5104	−0.6145	0.8305
−0.02	0.5018	−0.6138	0.8175
−0.03	0.5115	−0.6131	0.8342
−0.04	0.5121	−0.612	0.8366

TABLE III.

Investigating whether $T_{s} c = | U' (0) / V' (0) |$ corresponding to several suction and injection velocities.

T_s	$W^{*}$	$U' (0)$	$V' (0)$	$\| T_{s c} \|$
0.05	−0.55	0.4725	−0.9094	0.5195
0.05	−0.45	0.4825	−0.8509	0.5670
0.05	−0.35	0.4914	−0.7958	0.6174
0.05	0	0.5132	−0.6270	0.8185
0.05	0.35	0.5185	−0.4907	1.0566
0.05	0.452	0.5171	−0.4561	1.1337
0.05	0.55	0.5145	−0.4249	1.2108

T_s	$W^{*}$	$U' (0)$	$V' (0)$	$\| T_{s c} \|$
0.05	−0.55	0.4725	−0.9094	0.5195
0.05	−0.45	0.4825	−0.8509	0.5670
0.05	−0.35	0.4914	−0.7958	0.6174
0.05	0	0.5132	−0.6270	0.8185
0.05	0.35	0.5185	−0.4907	1.0566
0.05	0.452	0.5171	−0.4561	1.1337
0.05	0.55	0.5145	−0.4249	1.2108

Then, in terms of the lower deck coordinate

ξ = ζ / ε

⁠, we express the main deck stretched coordinate and expand the basic flow for small ζ, so we get

U = ε U_{0} ξ + ε^{2} U_{1} ξ^{2} + ε^{3} U_{2} ξ^{3} + \dots,

(49)

V = ε V_{0} ξ + ε^{2} V_{1} ξ^{2} + ε^{3} V_{2} ξ^{3} + \dots,

(50)

where

U_{j - 1} = U^{(j)} (0) / j!

and

V_{j - 1} = V^{(j)} (0) / j!

for

j = 1, 2, \dots

⁠.

We can see that fields of lower deck disturbance are expressed in the form

u = ε^{- 1} u_{- 1}^{L} (ξ) + u_{0}^{L} (ξ)

⁠, with a similar expression for v. However, as previously, for the pressure perturbations and surface-normal, we take

p = ε^{3} p_{0}^{L} (ξ) + ε^{4} p_{1}^{L} (ξ)

and

w = ε^{3} w_{0}^{L} (ξ) + ε^{4} w_{1}^{L} (ξ)

to be similar to the solution of the main deck. Taking these expansions and substituting the basic flow expansions and to match with the leading order terms of the main deck solutions written in terms of ξ. All this results in similar lower deck disturbance field expansions to those outlined in Hall³ and Hussain,⁷ namely,

u = \frac{r γ_{0} C}{ε {(β_{0} - α_{0} r T_{s})}^{2}} [U_{0} + 2 ε U_{1} ξ + \dots] + \frac{u_{- 1}^{L}}{ε} + u_{0}^{L} (ξ) + ε u_{1}^{L} (ξ) + \dots,

(51)

v = \frac{r γ_{0} C}{ε {(β_{0} - α_{0} r T_{s})}^{2}} [V_{0} + 2 ε V_{1} ξ + \dots] + \frac{v_{- 1}^{L}}{ε} + v_{0}^{L} (ξ) + ε v_{1}^{L} (ξ) + \dots,

(52)

\begin{matrix} w = - \frac{i γ_{0} ε^{5}}{{(β_{0} α_{0} r T_{s})}^{2}} [(α_{0} U_{1} r + β_{0} V_{1}) ξ^{2} + \\ ε (α_{0} r U_{2} + β_{0} V_{2}) ξ^{3} + \dots] + ε^{6} w_{1}^{L} (ξ), \end{matrix}

(53)

p = ε^{3} p_{1}^{L} (ξ) + \dots .

(54)

Substitution of these expansions into the disturbance equations and solving for (⁠

u_{- 1}^{L}, v_{- 1}^{L}

⁠), (⁠

u_{0}^{L}, v_{0}^{L}

⁠), and

(u_{1}^{L}, v_{1}^{L}, w_{1}^{L}, p_{1}^{L}

⁠) would help obtain the solutions in the lower deck. Through equating terms of

O (ε^{- 4})

and

O (ε^{- 5})

⁠, we get the relations

v_{- 1}^{L} (ξ) = - \frac{α_{0} r}{β_{0}} u_{- 1}^{L} (ξ),

(55)

v_{0}^{L} (ξ) = - \frac{α_{0} r}{β_{0}} u_{0}^{L} (ξ) .

(56)

Interestingly, from Eq. , we have

\frac{α_{0} r}{β_{0}} = - \frac{V' (0)}{U' (0)}

⁠. Next, we turn to the radial momentum disturbance equation, equating terms of

O (ε^{- 2})

and

O (ε^{- 3})

to give us the governing differential equation for

u_{- 1}^{L} (ξ)

⁠, as well as an equation for

u_{0}^{L} (ξ)

and

w_{1}^{L} (ξ)

⁠, giving

\frac{d^{2} u_{- 1}^{L}}{d ξ^{2}} - i (α_{0} r U_{1} + β_{0} V_{1}) ξ^{2} u_{- 1}^{L} = 0,

(57)

\begin{matrix} - i (α_{0} r U_{1} + β_{0} V_{1}) ξ^{2} u_{0}^{L} + \frac{d^{2} u_{0}^{L}}{d ξ^{2}} = r U_{0} w_{1}^{L} + i (α_{0} r U_{2} + β_{0} V_{2}) ξ^{3} u_{- 1}^{L} \\ + i (α_{0} r U_{0} + β_{1} V_{0}) ξ [\frac{r γ_{0} C U_{0}}{{(β_{0} α_{0} r T_{s})}^{2}} + u_{- 1}^{L}] . \end{matrix}

(58)

Regarding the boundary conditions from the no-slip condition at the wall and matching with the main deck away from the wall at

O (ε^{- 1})

and O(1) lead to

u_{- 1}^{L} = - \frac{r γ_{0} C U_{0}}{{(β_{0} - α_{0} r T_{s})}^{2}}, u_{0}^{L} = 0, on ξ = 0,

(59)

u_{- 1}^{L} \to 0, u_{0}^{L} \to \frac{r γ_{0} D U_{0}}{{(β_{0} - α_{0} r T_{s})}^{2}}, as ξ \to \infty,

(60)

for some constant D. Using substitution is

ρ = \sqrt{2} Δ^{1 / 4} ξ, where Δ = i (α_{0} r U_{1} + β_{0} V_{1}),

(61)

then we had to solve the parabolic cylinder equation for

u_{- 1}^{L}

⁠, given by

\frac{d^{2} u_{- 1}^{L}}{d ρ^{2}} - \frac{ρ^{2}}{4} u_{- 1}^{L} = 0.

(62)

To eliminate the constant of integration, we can use the boundary condition on ξ = 0. While to justify ignoring the exponentially growing solution,

V (0, ρ)

⁠, we use the condition as

ξ \to \infty

resulting in the following solution that is closely related to Hall's solution:³^,

u_{- 1}^{L} (ξ) = - \frac{U_{0} γ_{0} C r}{{(β_{0} - α_{0} T_{s})}^{2}} \frac{U (0, \sqrt 2 Δ^{1 / 4} ξ)}{U (0, 0)},

(63)

where

U (0, \sqrt{2} Δ^{1} ξ)

is a parabolic cylinder function (see Abramowitz and Stegun⁴⁰).

B. First-order lower-deck solution

1. Positive/negative strength of axial flow T_s

In this case, we must first solve for

w_{1}^{L} (ξ)

in order to find

u_{0}^{L} (ξ)

and

v_{0}^{L} (ξ)

⁠. This requires considering the next-order approximation to the disturbance equations similar to Hall.³ In the continuity equation at

O (ε^{- 3})

and in the surface-normal component of the linearized disturbance equations

O (ε^{- 6})

⁠, we get

\begin{matrix} i (α_{0} u_{1}^{L} + \frac{β_{0} v_{1}^{L}}{r}) + \frac{d w_{1}^{L}}{d ξ} = - \frac{i r γ_{0} C}{{(β_{0} - α_{0} r T_{s})}^{2}} (α_{1} U_{0} + \frac{β_{1} V_{0}}{r}) - i (α_{1} u_{- 1}^{L} + \frac{β_{1} v_{- 1}^{L}}{r}), \end{matrix}

(64)

\frac{d p_{1}^{L}}{d ξ} = 0 \Rightarrow p_{1}^{L} (ξ) = C .

(65)

Next, we aimed to expand the radial and azimuthal disturbance equations to

O (ε^{- 1})

⁠. Regarding the still fluid rotating disk, Hall³ observed that it is at this order the Coriolis effect terms from the choice of rotating co-ordinate system balance with the viscous terms. This is physically the case for the rotating cone boundary layer. These observations guide us to analyze the lower branch viscous mode. According to Hall's idea, we reduce the terms in W by the continuity equation expansion in terms of ε and noting that

W = O (ε^{2})

⁠. This does not play a role in the equations at

O (ε^{- 1})

⁠. Overall, these obtained equations are similar to those in Hall³^,

\begin{matrix} i u_{1}^{L} (α_{0} r U_{1} + β_{0} V_{1}) ξ^{2} + 2 r w_{1}^{L} U_{1} ξ - 2 (v_{- 1}^{L} + \frac{r γ_{0} C V_{0}}{{(β_{0} - α_{0} r T_{s})}^{2}}) \\ + i ξ (α_{1} U_{0} r + β_{1} V_{0}) (u_{0}^{L} + \frac{2 U_{1} ξ r γ_{0} C}{{(β_{0} - α_{0} r T_{s})}^{2}}) \\ + i ξ^{2} (α_{1} U_{1} r + β_{1} V_{1}) (u_{- 1}^{L} + \frac{U_{0} r γ_{0} C}{β_{0} - α_{0} r T_{s})^{2}}) \\ + i ξ (α_{2} U_{0} r + β_{2} V_{0}) (u_{- 1}^{L} + \frac{U_{0} r γ_{0} C}{{(β_{0} - α_{0} r T_{s})}^{2}}) \\ + i ξ^{3} (α_{0} r U_{2} + β_{0} V_{2}) u_{0}^{L} = - i α_{0} p_{1}^{L} + \frac{d^{2} u_{1}^{L}}{d ξ^{2}} + \frac{6 U_{2} r γ_{0} C}{{(β_{0} - α_{0} r T_{s})}^{2}}, \end{matrix}

(66)

\begin{matrix} i v_{1}^{L} (α_{0} r U_{1} + β_{0} V_{1}) ξ^{2} + 2 r w_{1}^{L} V_{1} ξ - 2 (u_{- 1}^{L} + \frac{r γ_{0} C U_{0}}{{(β_{0} - α_{0} r T_{s})}^{2}}) \\ + i ξ (α_{1} U_{0} r + β_{1} V_{0}) (v_{0}^{L} + \frac{2 V_{1} ξ r γ_{0} C}{{(β_{0} - α_{0} r T_{s})}^{2}}) \\ + i ξ^{2} (α_{1} U_{1} r + β_{1} V_{1}) (v_{- 1}^{L} + \frac{V_{0} r γ_{0} C}{(β_{0} - α_{0} r T_{s})^{2}}) \\ + i ξ (α_{2} U_{0} r + β_{2} V_{0}) (v_{- 1}^{L} + \frac{V_{0} r γ_{0} C}{{(β_{0} - α_{0} r T_{s})}^{2}}) \\ + i ξ^{3} (α_{0} r U_{2} + β_{0} V_{2}) v_{0}^{L} = - \frac{i β_{0} p_{1}^{L}}{r} + \frac{d^{2} v_{1}^{L}}{d ξ^{2}} + \frac{6 V_{2} r γ_{0} C}{{(β_{0} - α_{0} r T_{s})}^{2}} . \end{matrix}

(67)

To make the above expressions simpler, we perform the manipulation

i α_{0}

+

\frac{i β_{0}}{r}

. In addition, we expand the radial and azimuthal basic flow Eqs. and in powers of ε to get the values

U_{1} = \frac{U^{″} (0)}{2!} = - \frac{1 + T_{s}^{2}}{2},

(68)

V_{1} = \frac{V^{″} (0)}{2!} = 0,

(69)

as well as their derivatives expansion with regard to η to get the values

U_{2} = \frac{U^{″'} (0)}{3!} = - \frac{V_{0}}{3},

(70)

V_{2} = \frac{V^{″'} (0)}{3!} = \frac{U_{0}}{3},

(71)

which help in the Coriolis terms elimination in V₀ and U₀ in Eqs. and , respectively, resulting in an equation similar to that presented in Hall,³^,

\begin{matrix} i (α_{0} \frac{d^{2} u_{1}^{L}}{d ξ^{2}} + \frac{β_{0}}{r} \frac{d^{2} v_{1}^{L}}{d ξ^{2}}) + γ_{0}^{2} p_{1}^{L} + 2 i α_{0} v_{- 1}^{L} - \frac{2 i β_{0}}{r} u_{- 1}^{L} \\ = - ξ^{2} (α_{0} r U_{1} + β_{0} V_{1}) (α_{0} u_{1}^{L} + \frac{β_{0} v_{1}^{L}}{r}) + 2 i ξ (α_{0} r U_{1} + β_{0} V_{1}) w_{1}^{L} \\ - \frac{2 r γ_{0} C ξ^{2}}{{(β_{0} - α_{0} r T_{s})}^{2}} (α_{1} r U_{0} + β_{1} V_{0}) (α_{0} U_{1} + \frac{β_{0} V_{1}}{r}) . \end{matrix}

(72)

Importantly, the effects of Coriolis which survive are manifested by the terms in

u_{- 1}^{L}, v_{- 1}^{L}

⁠. At this order, this effect plays a significant role in balancing the viscosity effects. This indicates that the neutral curve structure for stationary small-wavenumber disturbances includes a dominant balance of both viscous and Coriolis terms. By the operation

\frac{d^{2}}{d ξ^{2}}

+ , we can eliminate

u_{1}^{L}

and

v_{1}^{L}

from the system. Upon such simplification and applying , , and , it is possible to obtain the governing equation for

w_{1}^{L}

which is similar to that in Hall,³ given by

\begin{matrix} \frac{d^{3} w_{1}^{L}}{d ξ^{3}} - i α_{0} r U_{1} ξ^{2} \frac{d w_{1}^{L}}{d ξ} + 2 i ξ α_{0} r U_{1} w_{1}^{L} \\ = γ_{0}^{L} C + (α_{1} r U_{0} + β_{1} V_{0}) \frac{r γ_{0} ξ^{2} C α_{0} U_{1}}{{(β_{0} - α_{0} r T_{s})}^{2}} \\ + \frac{2 i γ_{0} β_{0} C}{{(β_{0} - α_{0} r T_{s})}^{2}} (1 + \frac{V_{0}^{2}}{U_{0}^{2}}) U_{0} \frac{U (0, \sqrt{2} s)}{U (0, 0)}, \end{matrix}

(73)

where

s = Δ^{1 / 4} ξ

⁠. Solving this equation for

w_{1}^{L}

requires posing complementary function and particular integral form

w_{1}^{L} = w_{1}^{C . F .} + w_{1}^{P . I .} |

⁠, finding that

\begin{matrix} w_{1}^{L} = Δ^{- 3 / 4} (γ_{0}^{2} C F_{1} (s) + \frac{2 i γ_{0} β_{0} C}{{(β_{0} - α_{0} r T_{s})}^{2}} [1 + \frac{V_{0}^{2}}{U_{0}^{2}}] \frac{U_{0}}{U (0, 0)} F_{2} (s)) \\ + k_{1} ξ^{2} - i (α_{1} r U_{0} + β_{1} V_{0}) \frac{γ_{0} C}{{(β_{0} - α_{0} r T_{s})}^{2}} ξ, \end{matrix}

(74)

here

k_{1} =

const, and the independent solutions F₁ and F₂, presented by Hall,³ satisfy

F_{1}^{″'} - s^{2} F_{1}^{'} + 2 s F_{1} = 1, F_{1} (0) = F_{1} (\infty) = 0,

(75)

F_{2}^{″'} - s^{2} F_{2}^{'} + 2 s F_{2} = U (0, \sqrt{2} s), F_{2} (0) = F_{2} (\infty) = 0,

(76)

respectively. Collectively including this solution, we return to the continuity equation and substitute ξ = 0, using no-slip condition

u_{1}^{L} (0) = v_{1}^{L} (0) = 0

to obtain the eigen relation

\begin{matrix} γ_{0}^{2} I_{3} + \frac{i γ_{0} β_{0} U_{0}}{(β_{0} - α_{0} r T_{s})} (1 + \frac{V_{0}^{2}}{U_{0}^{2}}) I_{4} = \frac{i Δ^{1 / 2} γ_{0}}{{(β_{0} - α_{0} r T_{s})}^{2}} (α_{1} r U_{0} + β_{1} V_{0}), \end{matrix}

(77)

where

I_{3} = F_{1}^{'} (0)

and

I_{4} = \frac{2 F_{2}^{'} (0)}{U (0, 0)}

⁠, and in terms of definite integrals, they can be expressed involving parabolic cylinder functions. This is reached by applying a series of substitutions initially derived by Hall³ for the rotating disk in still fluid. Using the same way, Hussain⁷ used for rotating disk in axial flow

T_{s} > 0

⁠. Following their methods, the final aim has become transforming and so that they can be solved in terms of parabolic cylinder functions. For F₁, at first, we let

F_{1} = s^{2} G

⁠, which leads to

s_{2} G^{″'} + 6 s G^{″} + (6 - s^{4}) G' = 1.

(78)

So far, we suppose that if we let

H = G^{'}

⁠, this equation becomes

s_{2} H^{″} + 6 s H^{'} + (6 - s^{4}) H = 1.

(79)

Next, we change the variable

H = \frac{K}{s^{3}}

⁠, followed by

u = \sqrt{2} s

to obtain an analogous parabolic cylinder equation with inhomogeneous right-hand side, given by

K^{″} (u) - \frac{u^{4}}{4} K (u) = \frac{u}{2 \sqrt{2}} .

(80)

Regarding the parabolic cylinder function scale, we let

K (u) = R (u) U (0, u)

⁠, resulting in a first-order equation for

R^{'}

⁠, given by

R^{″} U + 2 R^{'} U^{'} = \frac{u}{2 \sqrt{2}},

(81)

which can be solved via using the integrating factors methods to get

R^{'} = \frac{1}{2 \sqrt{2}} \frac{\int_{\infty}^{u} u^{'} U (0, u^{'}) d u^{'}}{U {(0, u)}^{2}} .

(82)

Hence, re-expressing this solution back in terms of F₁, we obtain

u F^{'} (u) - 2 F_{1} (u) = R (u) U (0, u) .

(83)

Setting u = 0 and using

F_{1} |_{u = 0}

gives

R (0) = 0

⁠, as

U (0, 0) \neq 0

⁠. We subsequently differentiate and substitute u = 0, noting the change of variable back to s from u, to obtain

I_{3} = F_{1}^{'} |_{s = 0} = \frac{\int_{0}^{\infty} θ U (0, θ) d θ}{2 U (0, 0)} = 0.5984.

(84)

Carrying out a similar series of substitutions for F₂ considering the right-hand side difference resulting in the corresponding result for I₄, given by

I_{4} = \frac{2 F_{2}^{'} |_{s = 0}}{U (0, 0)} = \frac{\int_{0}^{\infty} θ U {(0, θ)}^{2} d θ}{U {(0, 0)}^{2}} = 0.4570.

(85)

The values of I₃, I₄ are estimated based on Simpson's rule with fixed steps of 0.1 from

θ = 0.0

to 5.0 using the same values that Hall³ and Hussain⁷ used for rotating disk. It is observable that

U (0, θ)

decays exponentially. Hence, we can solve making use of Eq. to get the leading order wavenumber and waveangle estimates in a similar way to the ones outlined by Hall³ for the rotating disk; at

T_{s} > 0.25

⁠, we obtained

\begin{matrix} γ_{0} = \frac{β_{0} U_{0}}{{(β_{0} - α_{0} r T_{s})}^{2}} (1 + \frac{V_{0}^{2}}{U_{0}^{2}}) \frac{I_{4}}{I_{3}} \\ = \frac{{(1 + \frac{V_{0}^{2}}{U_{0}^{2}})}^{3 / 4}}{| 1 + \frac{V_{0} T_{s}}{U_{0}} |} {(\frac{U_{0} I_{4}}{I_{3}})}^{1 / 2} r^{- 1 / 2} = 2.0603 r^{- 1 / 2}, \end{matrix}

(86)

\frac{α_{1}}{β_{0}} - \frac{β_{1} α_{0}}{β_{0}^{2}} = \frac{2 γ_{0}^{3 / 2} {(1 + \frac{V_{0}^{2}}{U_{0}^{2}})}^{- 1 / 4} {(1 + \frac{V_{0} T_{s}}{U_{0}})}^{2} I_{3}}{{| U_{0} V_{0} (1 + T_{s}^{2}) |}^{1 / 2} r^{1 / 2}} = 2.996 r^{- 5 / 4} .

(87)

where the numerical calculations are given for a rotating disk in positive axial flow

U_{0} = 0.6219, V_{0} = - 0.7359

at

T_{s} > 0.25

⁠. It is crucial noting the difficulty to find α₁ and β₁ independently. Therefore, it is a must to obtain

u_{0} (ξ)

or

v_{0} (ξ)

explicitly for the purposes of our analysis; they would not generate any additional useful information. Instead, we deflect our attention to the combination of α₁ and β₁ in terms of the waveangle

ϕ

between the radial vector and the normal to the spiral vortices. We have

\begin{matrix} tan (\frac{π}{2} - ϕ) = \frac{α r}{β} = \frac{(α_{0} + ε^{2} α_{1} + \dots) r}{(β_{0} + ε^{2} β_{1} + \dots)}, \\ = \frac{α_{0} r}{β_{0}} + ε^{2} (\frac{α_{1}}{β_{0}} - \frac{β_{1} α_{0}}{β_{0}^{2}}) r, \\ = 1.833 + 2.996 ε^{2} r^{- 1 / 4} . \end{matrix}

(88)

We note that the total wavenumber, scaled appropriately on the viscous mode wavelength, is given by

ε^{4} γ = 2.0603 ε^{4} r^{- 1 / 2} + \dots .

(89)

As previously mentioned regarding the inviscid modes, we follow Hall³ and identify the Reynolds number based on the thickness of boundary layer

δ^{*}

⁠, given by

R_{δ^{*}} = R^{- 1 / 2} r,

(90)

where the local wavenumber on the lower branch in this notation is given by

γ_{δ^{*}} = 2.0603 R_{δ^{*}}^{- 1 / 2} + \dots,

(91)

and expands the local viscous mode waveangle as

tan (\frac{π}{2} - ϕ) = 1.1833 + 2.996 R_{δ^{*}}^{- 1 / 4} + \dots .

(92)

By repeating the same methods for the positive axial flow, we obtained the negative axial flow $γ_{0} = - 1.2078, \frac{α_{1}}{β_{0}} - \frac{β_{1} α_{0}}{β_{0}^{2}} = 17.277$ and $tan (\frac{π}{2} - ϕ) = 1.20 402 + 17.277 ε^{2} r^{- 1 / 4} .$ at the $T_{s} < 0$ ⁠.

These expressions are independent of the radial location r in relation to displacement-thickness Reynolds number. Figures 3–6 illustrate the lower branches predictions of the asymptotic neutral wavenumber and waveangle regarding positive and negative strength axial flow. We reused semi-log and log-log scales for the waveangle and wavenumber plots, respectively. Similar to the inviscid branch,¹⁴ increasing the T_s stabilized the flow by reducing the available wavenumbers. Furthermore, in the case of the waveangle, increasing the T_s toward that of a rotating disk has the effect of increasing the waveangle. While reducing T_s by increasing the available wavenumber substantially destabilizes the flow. Reducing T_s toward a rotating disk reduces the waveangle from −0.01 to −0.02 and increases from −0.02 to −0.04 as shown in Fig. 6. Similar to Hussain²¹ in a rotating disk (⁠ $ϕ = 90 °$ ⁠), we estimate that increasing the T_s might increase the spiral vortices deviation angle from the radial vector because of two causes: (1) an increased rotational shear force on each vortex spiral and (2) the deviation of these vortices from the radial direction. In relation to energy transfer, vortices carry less kinetic energy in the radial direction. However, the pure rotational effect of the disk combines it with the kinetic energy in the azimuthal direction leading to a large deviation waveangle.

FIG. 3.

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Plot illustrating predictions of asymptotic neutral wavenumber $γ_{δ} *$ against $R_{δ} *$ for type II modes for $T_{s} = 0.30 - 0.65$ ⁠. Large T_s values move the plots up as shown by the arrow.

FIG. 4.

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Plot illustrating predictions of asymptotic neutral waveangle $ϕ$ against $R_{δ} *$ for type II modes for $T_{s} = 0.30 - 0.65$ ⁠. Larger T_s values move the plots up as shown by the arrow.

FIG. 5.

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Plot illustrating predictions of asymptotic neutral wavenumber $γ_{δ^{*}}$ against $R_{δ^{*}}$ for type II modes for $T_{s} = (- 0.01) - (- 0.04)$ ⁠. Smaller T_s values move the plots down as shown by the arrow.

FIG. 6.

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Plot illustrating predictions of asymptotic neutral waveangle $ϕ$ against $R_{δ^{*}}$ for type II modes for $T_{s} = (- 0.01) - (- 0.04)$ ⁠. Smaller T_s from −0.01 to −0.02 values move the plots down, while from −0.02 to −0.04 move the plot up as shown by the arrow.

2. Various positive/negative $W^{*}$ with fixed axial flow $T_{s} = 0.05$

Next, we investigated a rotating disk in fixed axial flow $T_{s} = 0.05$ through various suction/injection parameters to estimate the local wavenumber and waveangle for the type II lower-branch disturbance modes. Following the first-order analysis, we found that the integrals I₃ and I₄ remain the same as (84) and (85). We used Table III from Al Saeedi and Hussain paper¹⁴ and substituted into the viscous eigenrelation (77) to obtain the viscous mode type II wavenumber and waveangle values for various suction/injection parameters within fixed axial flow illustrated in Figs. 7 and 8. When we shifted from injection to suction, the wavenumber decreased and the curves vertically shifted downward as shown in Fig. 7 resulting in stabilizing the flow. While the waveangle shifted upward as shown in Fig. 8 increasing the distance between each waveangle value.

FIG. 7.

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Plot illustrating predictions of asymptotic neutral wavenumber predictions $γ_{δ^{*}}$ against $R_{δ^{*}}$ for type II modes for $T_{s} = 0.05$ with $W^{*} = (- 0.55)$ to $(0.55)$ ⁠. Increasing $W^{*}$ shifts the curves vertically downward.

FIG. 8.

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Plot illustrating predictions of asymptotic neutral waveangle predictions $ϕ$ against $R_{δ^{*}}$ for type II modes for $T_{s} = 0.05$ with $W^{*} = (- 0.55)$ to $(0.55)$ ⁠. Increasing $W^{*}$ shifts the curves vertically upward.

IV. DISCUSSION OF TYPE I/TYPE II NEUTRAL CURVES ASYMPTOTIC

Figures 9–12 show the wavenumber and waveangle asymptotic results in both inviscid and viscous modes, for a rotating disk $ϕ = 90 °$ ⁠, against displacement thickness Reynolds number for positive and negative strength axial flow. Through wall mode results and inviscid modes by Al Saeedi and Hussain,¹⁴ we noticed the asymptotic curves for the waveangle $ϕ$ are greater than the wavenumber $γ_{δ^{*}}$ ⁠. However, the integral values I₃ and I₄ remained same as in Eqs. (84) and (85). Combining measurements from Tables I and II (Ref. 14) and substituting into the inviscid and viscous eigenrelations Eqs. (80) and (77),¹⁴ we obtained the combined asymptotic local wavenumber and waveangle estimates for large Reynolds numbers as shown in Figs. 9–12.

FIG. 9.

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Plot illustrating predictions of asymptotic neutral wavenumber $γ_{δ^{*}}$ against $R_{δ^{*}}$ for type I and lower branch modes for $T_{s} = 0.30 - 0.65$ ⁠. Large T_s values move the plots up as shown by the arrow.

FIG. 10.

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Plot illustrating predictions of asymptotic neutral waveangle $ϕ$ against $R_{δ^{*}}$ for type I and lower branch modes for $T_{s} = 0.30 - 0.65$ ⁠. Larger T_s values move the plots up, as shown by the arrow.

FIG. 11.

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Plot illustrating predictions of asymptotic neutral wavenumber $γ_{δ^{*}}$ against $R_{δ^{*}}$ for type I and lower branch modes for $T_{s} = (- 0.01) - (- 0.04)$ ⁠. Smaller T_s values move the plot down.

FIG. 12.

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Plot illustrating predictions of asymptotic neutral waveangle $ϕ$ against $R_{δ^{*}}$ for type I and lower branch modes for $T_{s} = (- 0.01) - (- 0.04)$ ⁠. Smaller T_s values move the plots down for type I, while type II moves the plot down and then up, as shown by the arrow.

According to Al Saeedi and Hussain,¹⁴ the inviscid type I mode local wavenumber, we noticed that axial flow introduction at positive T_s shifted the curves vertically upward as Fig. 11 (Ref. 14) shows. In contrast, negative T_s as shown in Fig. 13 (Ref. 14) shifted the curves downward to instability.

FIG. 13.

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Plot illustrating predictions of asymptotic neutral wavenumber $γ_{δ^{*}}$ against $R_{δ^{*}}$ type I and lower branch modes for $T_{s} = 0.05$ with $W^{*} = (- 0.55)$ to $(0.55)$ ⁠. Increasing $W^{*}$ shifts the curves vertically downward.

We conclude that an axial flow alters the spiral vortex wavenumbers in an exponential manner. In the inviscid case, the main finding was to increase wavenumbers quantity where we observed the onset of an unstable mode. On the other hand, in type II modes, the main finding was to increase wavenumbers quantity, and we observed the onset of a stable mode. In summary, increasing T_s to a positive value destabilizes the type I modes (Fig. 11)¹⁴ and stabilizes the type II modes (Fig. 3) for a flow over a rotating disk. Oppositely, the negative T_s onset reduced the wavenumbers values leading to stabilization in the type I modes (Fig. 13)¹⁴ and destabilization in the type II modes (Fig. 5) for a flow over a rotating disk. In the case of positive T_s waveangle branches, the inviscid type I curve exhibits uniformly increasing waveangles with the introduction of positive axial flow (Fig. 12).¹⁴ The disk surface deflected the oncoming radial forced flow and contributes to increasing the azimuthal shear effect on the flow. Oppositely, the negative T_s reduced waveangles with the introduction of negative axial flow (Fig. 14).¹⁴ However, for type II viscous modes, the dominant effect on the waveangle estimates comes from the ratio of radial and azimuthal velocity gradients at the wall, U₀ and V₀. This is because the viscous stability analysis is related to the wall shear. The leading order waveangle increases gradually as positive T_s increases (Fig. 4). While negative, T_s initially decreases as T_s decreases. However, eventually, the ratio of azimuthal to radial wall shear increases closely back to its initial value. This resulted in increasing the waveangle curve back toward the neutral curve for $T_{s} = - 0.04$ (Fig. 6). Similar to Hussain et al.,³³ positive axial flow destabilizes the type I cross flow and stabilizes type II.

FIG. 14.

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Plot illustrating predictions of asymptotic neutral waveangle predictions $ϕ$ against $R_{δ^{*}}$ for type I and lower branch modes for $T_{s} = 0.05$ with $W^{*} = (- 0.55)$ to $(0.55)$ ⁠. Increasing $W^{*}$ shifts the curves vertically upward.

Next, we combined the neutral curve wavenumber and waveangle for the inviscid mode¹⁴ with the viscous mode in terms of suction/injection parameters within fixed axial flow $T_{s} = 0.05$ as demonstrated in Figs. 13 and 14. From our recent study about the inviscid mode,¹⁴ we noticed that the introduction of fixed axial flow within suction and injection shifted the curves vertically downward from $w^{*} = (- 0.55) - (0.55)$ (Fig. 15).¹⁴ Similar to the case of T_s positive and negative, applying suction and injection altered the spiral vortex wavenumbers in an exponential manner. The main result is to reduce the quantity of wavenumbers where the onset of a stable mode is noticed at the smallest $W^{*}$ ⁠. In contrast, instability is increased in type II modes (Fig. 7). We concluded at $T_{s} = 0.05$ by increasing $W^{*}$ from negative to positive values shifted the curves vertically downward, stabilizing and destabilizing the flow at type I¹⁴ and type II modes, respectively, for a rotating disk (Figs. 13 and 14). In waveangle branches, the inviscid type I¹⁴ curve increased waveangles and enlarged vortices by gradually increasing $W^{*}$ from negative to positive as shown in Fig. 16.¹⁴ Moreover, the radial forced flow is deflected by the disk surface and increases the azimuthal shear effect on the flow onset (see Al Saeedi and Hussain¹⁴). In type II viscous modes, the dominant effect on the waveangle came from the ratio between radial U₀ and azimuthal velocity V₀ gradients at the wall. The order of waveangle initially increased as $W^{*}$ increased from negative to positive and vice versa (Fig. 8). Collectively, increasing $W^{*}$ from negative to positive shifted the curves at types I and II upward (Fig. 14). Our data are consistent with Al-Malki⁴¹ for Blasius flow, where suction stabilized the type I cross flow instability mode.

FIG. 15.

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Neutral curve of the wavenumber in the radial direction at positive T_s. The larger lobe is the type I mode with $R_{c} (I)$ ⁠, while the smaller lobe is the type II mode with $R_{c} (I I)$ ⁠.

FIG. 16.

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Neutral curve of the wavenumber in the azimuthal direction at positive T_s in the $(R_{c}, β)$ plane.

Importantly, expanding the inviscid type I and viscous type II modes provided the capacity to consider non-parallel effects. As Hall's work in the rotating disk,³ the inviscid type I mode asymptotic expansion included viscous effects. However, they disappeared in the leading and next order corrections and would reappear if we expanded the asymptotic analysis in inviscid mode. Similarly, type II mode (viscous), using the triple-deck theory in the rotating disk, would enable us to consider non-parallel effects.³ Smith³⁸ also considered these effects in the Blasius boundary layer when using a triple-deck structure.

V. NEUTRAL CURVES

We used the Chebyshev–Tau spectral method to obtain the neutral stability solutions. Here, the exponential map helped us transform the Gauss–Lobatto grid points into the physical domain. After that, we focused on solving linear stability equations as variables primitive more than 100 of the distributed grid points between the geometry of the disk surface η = 0 and the domain of top $η = 20_{maxi}$ ⁠. Hence, this enabled us to differentiate between the type I (upper) and type II (lower) critical Reynolds number boundaries and respective lobes of the neutral curves. The code used here is adapted from Al-Malki et al.⁴³

We computed the neutral stability curves for various parameters combining the mean flow for a rotating disk in axial flow with various suction/injection parameters $W^{*}$ and positive/negative strengths of axial flow T_s. Through a neutral curve, we can visualize the boundary between decay and exponential growth of any perturbation in the parameter space. We concluded that the interior part of the neutral curve, containing the smallest parameter values, leads to an instability where ‘tip’ is the critical number (see Miller et al.³⁹). The boundary of the curve is the parameter values that have neutral stability at $α_{i} = 0$ in the $(R_{c}, β)$ and $(R_{c}, n)$ planes. $ϕ = {tan}^{- 1} (β / α_{r})$ is the direction angle of spiral vortices related to circle concentric to the disk. Therefore, n and $ϕ$ are physical and measurable quantities. n means an integer taking any value during the mathematical analysis.

To calculate the neutral curve of the system (12)–(15), the Reynolds number $\bar{R_{ε}}$ is set and the wavenumber β in the azumithal direction is iterated. We solved the quadratic eigenvalue problem at each wavenumber and organized the spectrum of α in the radial direction according to their size with a min $| α_{i} |$ chosen as the branch point. The iterative loop continues until $α_{i} = 0$ ⁠, in which these values are recorded. This process can be repeated for another Reynolds number. This continues until the neutral curve is mapped out for a wide range of Reynolds numbers.

As a result, most of the neutral curves had a two-lobed structure: the type I crossflow and viscous instabilities represented in the upper and lower branches, respectively. This is consistent with all previous values of the boundary layer for the rotating disk. Therefore, this discussion focuses on the critical Reynolds number, $R_{c} (I)$ type I and $R_{c} (I I)$ type II (the lowest Reynolds number that permits an unstable mode). Hence, increasing the Reynolds number from the critical Reynolds number for a fixed parameter will make the flow unstable, and vice versa for reducing the Reynolds number.

A. Axial flow (no suction/injection)

In this case, the effect of adding axial flow to the perturbation equations (12)–(15) will lead us to changes in the mean flow. Figures 15–17 show the stationary mode neutral curves for a range of strengths axial flow T_s positive. In positive T_s, we observed that as T_s augments, the neutral curves directed to the right toward higher Reynolds numbers, see Tables IV and V. Therefore, the critical Reynolds number is also augmented. This increase implies that the instability region is shifted toward higher Reynolds numbers, while the stability region is enlarged. In addition, as T_s increases, the type II will become significantly relevant to the type I. This indicates that the type II modes became more noticeable than before. Moreover, increasing T_s moves the neutral curve upper branch to cover more wavenumbers in the radial direction α shown in Fig. 15. As a result, in both type lobes I and II, α (wavenumber in the radial direction) increases with increasing T_s. This would suggest that the critical Reynolds numbers in the instability ranges increase with regard to the latitudinal wavenumber as well as in the azimuthal direction β shown in Fig. 16 in both types I and II. Furthermore, the effective wavenmber n also increases significantly with increasing T_s, see Fig. 17. The critical wave angle also increased in a similar manner to that of n in Fig. 18. Finally, the range of unstable wave angles is diminished as T_s is increased even though the curve's unstable angles are significantly increased.

FIG. 17.

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Neutral curve of the effective wavenumber at positive T_s in the $(R_{c}, n)$ plane.

TABLE IV.

The critical values for the neutral curve with positive axial flow T_s on the upper-branch for stationary waves.

T_s	R_c	α_r	β	n	$ϕ$
0.30	535.80	0.556	0.227	121.7	22.20
0.35	596.90	0.576	0.260	155.5	24.29

T_s	R_c	α_r	β	n	$ϕ$
0.30	535.80	0.556	0.227	121.7	22.20
0.35	596.90	0.576	0.260	155.5	24.29

TABLE V.

The critical values for the neutral curve with positive axial flow T_s on the lower branch for stationary waves.

T_s	R_c	α_r	β	n	$ϕ$
0.30	559.60	0.2	0.106	54.2	27.92
0.35	604.80	0.214	0.118	71.6	28.87

T_s	R_c	α_r	β	n	$ϕ$
0.30	559.60	0.2	0.106	54.2	27.92
0.35	604.80	0.214	0.118	71.6	28.87

FIG. 18.

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Angle neutral curve at positive T_s in the $(R_{c}, ϕ)$ plane.

On the other hand, Figs. 19–22 show the stationary mode neutral curves for a range of decreased axial flow T_s at negative values. Here, we noticed that as T_s decreased, the neutral curves were directed to the left toward lower Reynolds numbers. As a result, in Table VI, the critical Reynolds numbers decreased by reducing the T_s number from −0.01 to −0.03, while they increased at $T_{s} = - 0.04$ ⁠. This implies that the instability region is shifted toward lower Reynolds numbers, while the stable region is reduced. Therefore, decreasing T_s has a destabilizing effect in contrast to increasing T_s. We observed the effective wavenumber n at the critical Reynolds numbers decreases from 21.3 to 19.6 with decreasing T_s from −0.01 to −0.03, respectively. Then, at $T_{s} = - 0.04$ ⁠, it became constant at 19.6 value as shown in Fig. 21. Regarding the wavenumber in the azimuthal direction β, it decreased from 0.075 to 0.07 when reducing the T_s from −0.01 to −0.03. Then, at $T_{s} = - 0.04$ ⁠, it became constant as well at 0.07 value as illustrated in Fig. 21. Moreover, reducing T_s moves the neutral curve upper branch to cover more wavenumbers into radial direction α shown in Fig. 19. The critical wave angle $ϕ$ followed the behavior of the n as noted in Fig. 22. We observed that wave angle $ϕ$ decreased (⁠ $11.16 - 10.79$ ⁠) with decreasing T_s (⁠ $(- 0.01) - (- 0.03)$ ⁠); from T_s −0.04, it remained constant at 10.79 value as shown in Table VI. All curves are shifted to the left and the range of instability increased.

FIG. 19.

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Neutral curve of the wavenumber in the radial direction at negative T_s. The larger lobe is the type I mode with $R_{c} (I)$ ⁠, while the smaller lobe is the type II mode with $R_{c} (I I)$ ⁠.

FIG. 20.

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Neutral curve of the wavenumber in the azimuthal direction at negative T_s in the $(R_{c}, β)$ plane.

FIG. 21.

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Neutral curve of the effective wavenumber at negative T_s in the $(R_{c}, n)$ plane.

FIG. 22.

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Angle neutral curve at negative T_s in the $(R_{c}, ϕ)$ plane.

TABLE VI.

The critical values for the neutral curve with negative axial flow T_s on the upper branch for stationary waves.

T_s	R_c	α_r	β	n	$ϕ$
−0.04	281.30	0.367	0.070	19.6	10.79
−0.03	280.15	0.367	0.070	19.6	10.79
−0.02	282.19	0.374	0.073	20.6	11.04
−0.01	283.69	0.380	0.075	21.3	11.16

T_s	R_c	α_r	β	n	$ϕ$
−0.04	281.30	0.367	0.070	19.6	10.79
−0.03	280.15	0.367	0.070	19.6	10.79
−0.02	282.19	0.374	0.073	20.6	11.04
−0.01	283.69	0.380	0.075	21.3	11.16

B. Various positive/negative $W^{*}$ with fixed axial flow $T_{s} = 0.05$

Figures 23–26 show the neutral-stability curves for stationary $α_{i} = 0$ waves representing a range of suction/injection $W^{*}$ with positive fixed axial flow T_s. The suction case resulted in a more stabilizing effect, while the injection gave more unstable effects. However, we observed that small injection (⁠ $W^{*}$ = 0.55) was needed to increase the critical Reynolds number on both the upper and lower branches from 490.40 to 658.10 and from 579.90 to 705.10, respectively.

FIG. 23.

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Neutral curve of the wavenumber in the radial direction at fixed axial flow $T_{s} = 0.05$ within suction/injection $W^{*}$ ⁠. The larger lobe is the type I mode with $R_{c} (I)$ ⁠, while the smaller lobe is the Type II mode with $R_{c} (I I)$ ⁠.

FIG. 24.

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Neutral curve of the wavenumber in the azimuthal direction at fixed axial flow $T_{s} = 0.05$ within suction/injection $W^{*}$ in the $(R_{c}, β)$ plane.

FIG. 25.

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Neutral curve of the effective wavenumber at fixed axial flow $T_{s} = 0.05$ within suction/injection $W^{*}$ in the $(R_{c}, n)$ plane.

FIG. 26.

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Angle neutral curve at fixed axial flow $T_{s} = 0.05$ within suction/injection $W^{*}$ in the $(R_{c}, ϕ)$ plane.

According to Malik¹⁰ on the neutral curve, the effect of the suction made the lower lobes more clear. Therefore, the importance of the related lower wavenumber disturbance increased with increasing the suction rate. The ratio of the Reynolds number corresponding to the lower lobes (see Tables VII and VIII) to the critical Reynolds number increased. The values began approximately at 0.18 for $W^{*} = 0.35$ and 0.19 for $W^{*} = 0.55$ ⁠, as the suction rate increased. Hence, the lower lobes became dominant for the large suction values. That moved the position of the critical layer toward the wall with an increasing suction rate. However, we observed both type I and II modes wavenumber in the radial direction α increased with increasing the $W^{*}$ shown in Fig. 23. Figure 24 shows the wavenumber in the azimuthal direction β reduction with increased $W^{*}$ ⁠. In contrast, the effective wavenumber n in both lobes increased with increasing $W^{*}$ as shown in Fig. 25.

TABLE VII.

The critical values for the neutral curve suction with positive fixed axial flow $T_{s} = 0.05$ on the upper branch for stationary waves.

$W^{*}$	R_c	α_r	β	n	$ϕ$
0	307.10	0.417	0.094	28.9	12.73
0.35	409.40	0.472	0.089	43.8	10.67
0.45	566.80	0.491	0.088	49.9	10.16
0.55	658.10	0.511	0.087	57.9	9.66

$W^{*}$	R_c	α_r	β	n	$ϕ$
0	307.10	0.417	0.094	28.9	12.73
0.35	409.40	0.472	0.089	43.8	10.67
0.45	566.80	0.491	0.088	49.9	10.16
0.55	658.10	0.511	0.087	57.9	9.66

TABLE VIII.

The critical values for the neutral curve suction with positive fixed axial flow $T_{s} = 0.05$ on the lower branch for stationary waves.

$W^{*}$	R_c	α_r	Β	n	$ϕ$
0	441.60	0.147	0.054	23.8	20.17
0.35	576.90	0.167	0.051	29.6	16.98
0.45	634.80	0.175	0.050	32.0	15.94
0.55	705.10	0.183	0.049	34.8	14.98

$W^{*}$	R_c	α_r	Β	n	$ϕ$
0	441.60	0.147	0.054	23.8	20.17
0.35	576.90	0.167	0.051	29.6	16.98
0.45	634.80	0.175	0.050	32.0	15.94
0.55	705.10	0.183	0.049	34.8	14.98

Meanwhile, the waveangle $ϕ$ corresponding to the critical Reynolds number reduced with increasing the $W^{*}$ ⁠. However, the lower part of the neutral curve abscissa suggested why the waveangle predicted from inviscid theory.⁸ aligned with experimental results. However, the analysis of the inviscid mode could not predict perfectly the wavenumber or the number of spiral vortices expected to appear during the transition. We estimated the effect of suction on the predicted number of vortices by taking a value of the wavenumber at the critical Reynolds numbers.

TABLE IX.

The critical values for the neutral curve injection with positive fixed axial flow $T_{s} = 0.05$ on the upper branch for stationary waves.

$W^{*}$	R_c	α_r	β	n	$ϕ$
−0.55	165.97	0.351	0.100	16.6	15.90
−0.45	183.53	0.370	0.099	18.2	14.97
−0.35	204.16	0.379	0.098	20.0	14.49

$W^{*}$	R_c	α_r	β	n	$ϕ$
−0.55	165.97	0.351	0.100	16.6	15.90
−0.45	183.53	0.370	0.099	18.2	14.97
−0.35	204.16	0.379	0.098	20.0	14.49

Unlike the suction, the injection case had a destabilizing effect where it decreased the critical Reynolds numbers (Table IX). The waveangle $ϕ$ at the critical Reynolds numbers increases with decreasing $W^{*}$ as shown in Fig. 26 while the wavenumber in the radial direction decreased (23). We also observed the wavenumbers in the azimuthal direction β increased with decreasing the $W^{*}$ as shown in Fig. 24. On the contrary, the effective wave number n decreased, see Fig. 25. Hence, the injection case is more sensitive and clearly shows us type I inviscid mode due to crossflow instability behaves unlike the suction case. Conversely, an experiment with fluid injection would need strong forcing for subcritical instability despite the linear destabilizing of stationary type II waves due to injection. Overall, moving from injection to suction shifted the curves to the right and diminished the instability range. Type II started to appear as moving toward the suction.

VI. TYPES I AND II ASYMPTOTIC AND NUMERICAL COMPARISON

To compare the numerical and asymptotic results with regard to positive/negative axial flow and suction/injection with a fixed axial flow, we must create consistent comparisons for both types I and II. This will be achieved first by using the asymptotic method of waveangle between the radial vector and the normal to the spiral vortices, as well as the wavenumber as shown in Eqs. (91) and (92) for type II and Eqs. (81) and (82),¹⁴ for type I and, second, using the local Reynolds number defined in Sec. VI as numerical results for both branches. Meanwhile, we have to consider the instability to compare the asymptotic values of the α, β, and n because we are unable to separate both waveangle (⁠ $ϕ$ ⁠) and wavenumber (⁠ $γ_{δ^{*}}$ ⁠). Here, the numerical wavenumber k is equivalent to the asymptotic wavenumber of $γ_{δ^{*}}$ ⁠, and the use of k is done to be consistent with other comparison studies that used the numerical k wavenumber notation instead of the asymptotic wavenumber one. We could only compare the neutral curve results for waveangle and wavenumber types I and II. We observed, as shown in Figs. 27–38, that waveangle and wavenumber, respectively, the numerical results are halted at $R_{L} = 10^{4}$ for both types I and II because destabilization occurred beyond this point.

FIG. 27.

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Positive T_s, the solid line shows the numerical solutions of the neutral stability curves for the wavenumber. Blue and red lines represent the asymptotic solutions for types I and II, respectively.

FIG. 28.

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Positive T_s, the solid line shows the numerical solutions of the neutral stability curves for the waveangle. Blue and red lines represent the asymptotic solutions for types I and II, respectively.

FIG. 29.

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Negative T_s, solid line shows the numerical solutions of the neutral stability curves for the wavenumber. Blue and red lines represent the asymptotic solutions for types I and II, respectively.

FIG. 30.

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Negative T_s, the solid line shows the numerical solutions of the neutral stability curves for the waveangle. Blue and red lines represent the asymptotic solutions for types I and II, respectively.

FIG. 31.

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$W^{*} = 0.55, 0.45$ ⁠, the solid line shows the numerical solutions of the neutral stability curves for the wavenumber. Blue and red lines represent the asymptotic solutions for types I and II, respectively.

FIG. 32.

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$W^{*} = 0.35, 0$ ⁠, the solid line shows the numerical solutions of the neutral stability curves for the wavenumber. Blue and red lines represent the asymptotic solutions for types I and II, respectively.

FIG. 33.

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$W^{*} = - 0.35, - 0.45$ ⁠, the solid line shows the numerical solutions of the neutral stability curves for the wavenumber. Blue and red lines represent the asymptotic solutions for types I and II, respectively.

FIG. 34.

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$W^{*} = - 0.55$ ⁠, the solid line shows the numerical solutions of the neutral stability curves for the wavenumber. Blue and red lines represent the asymptotic solutions for types I and II, respectively.

FIG. 35.

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$W^{*} = 0.55, 0.45$ ⁠, the solid line shows the numerical solutions of the neutral stability curves for the waveangle. Blue and red lines represent the asymptotic solutions for types I and II, respectively.

FIG. 36.

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$W^{*} = 0.35, 0$ ⁠, the solid line shows the numerical solutions of the neutral stability curves for the waveangle. Blue and red lines represent the asymptotic solutions for types I and II, respectively.

FIG. 37.

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$W^{*} = - 0.35, - 0.45$ ⁠, solid line shows the numerical solutions of the neutral stability curves for the wavenumber. Blue and red lines represent the asymptotic solutions for types I and II, respectively.

FIG. 38.

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$W^{*} = - 0.55$ ⁠, solid line shows the numerical solutions of the neutral stability curves for the wavenumber. Blue and red lines represent the asymptotic solutions for types I and II, respectively.

From Fig. 27, we observed that the numerical results for type I wavenumber at $T_{s} = 0.3$ and 0.35 are not close enough to the type I asymptotic results. On the other hand, type II is in close agreement starting from $R_{L} = 10^{3}$ ⁠, after which it meets the asymptotic but continues to be below it as it gets closer to where the numerical is halted at 10⁴. Due to the numerical code, it is unclear whether this result would continue in similar patterns at higher Reynolds numbers. Regarding the waveangle at $T_{s} = 0.3$ and 0.35, as shown in Fig. 28, we observed that type I is much closer to agree with the numerical results compared to type II is not likely to agree. Similar to the wavenumer code, it is not clear to us how the results would be at higher R_L values.

For $T_{s} = - 0.01$ and −0.04, as shown in Fig. 29, the wavenumber results for type I looks close to each other at both T_s starting from $R_{L} = 10^{3}$ and getting closer after a few thousand onward until it is halted at $R_{L} = 10^{4}$ ⁠. While the type II asymptotic results are a bit different, type II are in close agreement from $R_{L} = 10^{3}$ after which it meets the asymptotic but continues to be below it as it gets closer to where the numerical is halted at $R_{L} = 10^{4}$ ⁠. Here, we speculate that the numerical results will continue below the asymptotic at higher R_L values. For waveangle at $T_{s} = - 0.01$ and −0.04 as shown in Fig. 30, type II of asymptotic results is very far from being agreeing with the numerical values, while type I is much closer to being agreed to start from 10³ and getting more closer after a few thousand reaching the 10⁴. After $R_{L} = 10^{4}$ ⁠, it is not obvious whether it will continue in the same way due to the numerical code limitation.

For the case of suction and injection with the fixed $T_{s} = 0.05$ as shown in Figs. 31–34, we observed that at $W^{*} = 0.55, 0.45, 0.35, - 0.35, - 0.45, - 0.55$ the wavenumber numerical results for type I agree with the asymptotic results where Reynolds number value reaching at 10⁴, except at $W^{*} = 0$ where the numerical solutions halted at 10⁴. We were unable to see whether the asymptotic and numerical results would agree after this point due to the numerical code for the branch I destabilized at $R = 10^{4}$ ⁠.

Regarding the type II numerical results, at $W^{*} = 0.55, W^{*} = 0.45$ ⁠, and $W^{*} = 0.35$ ⁠, we observed that it is close to agreeing with the asymptotic solutions at $R = 10^{4}$ ⁠, whereas at $W^{*} = 0, - 0.35, - 0.45, - 0.55$ ⁠, we speculated that the numerical and asymptotic solutions would agree at some point after $R = 10^{4}$ ⁠. Similarly, due to the destabilization type II asymptotic and numerical results, we were unable to determine their further agreement after $R = 10^{4}$ ⁠. Figures 35–38 show the waveangle with regard to the suction and injection numerical results for type I; we noticed that type I numerical results almost agree with the asymptotic results at $R_{L} = 10^{4}$ ⁠, except at $W^{*} = - 0.55$ ⁠. We also observed that at $W^{*} = 0.55$ until $W^{*} = - 0.55$ type II was not close with the asymptotic results; its results get more away while moving toward $W^{*} = - 0.55$ where both results are not in agreement at this point.

VII. CRITICAL REYNOLDS NUMBERS

In this section, we studied the effect of both suction and injection $W^{*}$ within various axial flows T_s for type I on the location of the critical Reynolds number. We have to take into account that the location of the critical Reynolds number for the type I mode is the most dangerous in terms of stability. That means that using the lobe with the lowest Reynolds number can result in a representative neutral curve estimate. To compare the effects of suction and injection modes within various axial flows for the branch I mode, we need to plot the Critical Reynolds number for each value of the axial flow T_s from 0.05 to 0.25 against increasing $W^{*}$ values of suction and injection from $W^{*} =$ −0.55 to 0.55 as shown in Figs. 39 and 40. From Figs. 39 and 40, we observed that increasing the axial flow results in an elevated critical Reynolds number. We also observed that moving from injection/negative (where the lowest critical Reynolds number exists) to suction/positive (where a higher critical Reynolds number exists) increased the stability. In other words, increasing $w^{*}$ along with T_s leads to more stabilization. Therefore, we suggest that the suction effect within an axial flow can result in a stronger stabilization (Fig. 40). As shown in Fig. 40, the combined effect of both suction and axial flow parameters appears to exhibit a more powerful controlling effect on the crossflow instability than just one of the parameters alone.

FIG. 39.

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R_c critical Reynolds number for each suction and injection $w^{*}$ within various T_s between 0.05 and 0.25.

FIG. 40.

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3D R_c critical Reynolds number for each suction and injection $w^{*}$ within various T_s between 0.05 and 0.25.

VIII. CONCLUSIONS

Through this study, we evaluated type II modes of a rotating disk by using an asymptotic method. We found that when T_s is increased, the flow became more stable as a result of decreasing and increasing the wavenumbers and waveangle, respectively, and vice versa, which is consistent with the physical interpretation of positive axial flow stabilizing according to Hussain,⁷ Hussain et al.,³³ and Al Malki.⁴² We also observed that shifting from injection to suction stabilized the flow. However, our results are consistent with Al-Malki⁴¹ for Blasius flow, in that suction causes a stabilization of the type II cross-flow instability mode. When we compared the asymptotic results of type I with II modes, we concluded that increasing T_s to a positive value destabilizes the type I modes and stabilizes the type II modes for a flow over a rotating disk and vice versa. We also concluded that at $T_{s} = 0.05$ ⁠, increasing $W^{*}$ from negative to positive values shifted the curves vertically downward, leading to stabilization and destabilization of the flow at type I and type II modes, respectively, for a rotating disk. In the neutral curves, we observed that increasing T_s directed the neutral curves to the right toward higher Reynolds numbers. Therefore, the instability region is shifted toward higher Reynolds numbers causing the stability region to enlarge, and vice versa when T_s is decreased. This implies that the instability region is shifted toward lower Reynolds numbers while the stable region is reduced. Therefore, decreasing T_s has a destabilizing effect in contrast to increasing T_s. However, our results are consistent with Miller et al.³⁹

Finally, we compared the asymptotic and numeric results for both types I and II followed by the critical Reynolds numbers comparisons. We observed that waveangle and wavenumber, the numerical results are halted at $R_{L} = 10^{4}$ for the positive and negative T_s and also halted at $R_{L} = 10^{4}$ for the suction/injection for both types I and II because destabilization occurred beyond this point. Regarding the wavenumber both type I and II, the numerical results almost agree with the asymptotic results. While the waveangle types I and II, some of the numerical results close to agreeing with the asymptotic results. Due to the limitations of the numerical code, it is unclear whether this result will continue in similar patterns at higher Reynolds numbers. However, the numerical code shows that for certain parameter ranges only, for the vast majority of parameters, the agreement between numerics and asymptotics is favorable and also agrees well for the relevant parameters with related rotating disk studies.³⁹ In general, the combined effect of both suction and axial flow parameters appears to exhibit a more powerful controlling effect on the crossflow instability than just one of the parameters alone.

ACKNOWLEDGMENTS

B. Al Saeedi would like to express his sincere thanks to Moataz Badawi and M. A. S. Al-Malike for their help in improving this paper.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Bashar Al Saeedi: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Fathy gamal Abdelrazik: Investigation (equal). Matthew Robert Fildes: Investigation (supporting); Writing – review & editing (supporting). Zahir Hussain: Supervision (lead); Visualization (supporting); Writing – review & editing (equal).

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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I. INTRODUCTION

II. PROBLEM FORMULATION

III. ASYMPTOTIC LINEAR DISTURBANCE EQUATIONS

A. Leading order triple-deck solution

B. First-order lower-deck solution

1. Positive/negative strength of axial flow Ts

2. Various positive/negative W * with fixed axial flow T s = 0.05

IV. DISCUSSION OF TYPE I/TYPE II NEUTRAL CURVES ASYMPTOTIC

V. NEUTRAL CURVES

A. Axial flow (no suction/injection)

B. Various positive/negative W * with fixed axial flow T s = 0.05

VI. TYPES I AND II ASYMPTOTIC AND NUMERICAL COMPARISON

VII. CRITICAL REYNOLDS NUMBERS

VIII. CONCLUSIONS

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

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1. Positive/negative strength of axial flow T_s

2. Various positive/negative $W^{*}$ with fixed axial flow $T_{s} = 0.05$

B. Various positive/negative $W^{*}$ with fixed axial flow $T_{s} = 0.05$