1 Introduction

The supersonic flow around a sharp corner is one of the most important elementary flows, which is affected by a simple wave. One interesting question is that how the flow turns the corner. For steady flow around a convex corner, when the oncoming flow arrives with a constant velocity a long a wall, it will turn the corner by a centered expansion simple wave or a centered compression simple wave and continue along the rigid wall by another constant state^[1]. For the concave case, if the angle of the corner is less than a critical value, there may appear a strong shock (transonic shock) or a weak shock (transonic shock or supersonic shock). As indicated in Ref. [1], the weak shock is stable but the stability of the strong shock is not clear^[2-9]. The research on the flow passing the corner is mostly on the steady flow. However, the research results on unsteady flow are few. Sheng and You^[10] constructed the self-similar solution of the expansion problem arising for the phenomenon that two-dimensional (2D) pseudo-steady isentropic irrotational supersonic flow turned a sharp corner and expanded into vacuum, and proved that the supersonic flow turned the convex corner by a centered expansion wave or a centered compression wave^[11].

In this paper, we will study the 2D pseudo-steady isentropic isothermal flow around the convex corner. The flow is described by the following compressible Euler equations:

(1)

where (u, v), ρ, and p represent the velocity, the density, and the pressure, respectively, and the state equation is p=ρ.

As shown in Fig. 1(a), a convex corner at the point O is formed by the horizontal straight rigid wall AO and the sloping straight rigid wall OB with an inclination angle -θ (0 < θ < π). The supersonic oncoming flow is at the constant state (u₁, 0, ρ₁) in a region adjacent to the straight wall AO before O. The problem is how the oncoming flow turns the corner near the point O locally to arrive at a appropriate state (u₂, v₂, ρ₂) such that v₂/u₂=tan(-θ). Figure 1(b) shows the fluid flows under the self-similar transformations

The main results of this paper, which we will prove constructively, are given in the following theorem.

Theorem 1   For the supersonic oncoming flow (u₁, 0, ρ₁), if the density of the downstream flow is less than ρ₁, the flow will turn the corner near the point O locally by an incomplete centered expansion simple wave and a constant state. If the density of the downstream flow is greater than ρ₁ and the oncoming flow will turn the corner by a simple centered compression wave and a constant state, where β₂^* is the critical angle that can form a simple complete centered compression wave.

The paper is organized as follows. In Section 2, we give the characteristic analysis of the 2D isentropic irrotational pseudo-steady Euler equations for isothermal flow. In Section 3, we discuss the properties of the centered simple waves by the definition of the principal part of the centered wave. In Section 4, we concern the structure of the centered simple waves deduced from the supersonic flow around a convex corner.

2 Pseudo-steady Euler equations and characteristic analysis

For smooth flow, Eq. (1) can be written for isothermal flow with the self-similar variables (ξ, η) as follows:

(2)

where (U, V)=(u-ξ, v-η) is called the pseudo-flow velocity. Assuming that the flow is irrotational, i.e., u_y=v_x, from the last two equations of the system (2), the pseudo-Bernoulli law is obtained as follows:

(3)

where φ(ξ, η) is a potential function introduced by φ_ξ=U and φ_η=V. Therefore, the system (2) can be governed by Eq. (3), and

(4)

The eigenvalues of Eq. (2) are determined by

(5)

which gives

Obviously, if and only if U²+V²>1, the system (4) is hyperbolic. The pseudo-wave characteristics C_± are defined as the integral curves of

The pseudo-stream curve C₀ is defined as the integral curves of

The left-eigenvectors of the eigenvalues λ_± are

Multiplying Eq. (4) by l_± on the left and differentiating Eq. (3), it is easy to get the characteristic equations of the system (2) as follows:

(6)

where

By introducing the characteristic inclination angle variables α, β, and σ (see Refs. [12]-[15]), u, v, and λ_± can be expressed as follows (see Fig. 2):

where σ and δ are the inclination angles of the pseudo-flow characteristic and the pseudo-Mach angle, respectively.

3 Centered simple wave

In this section, we will construct the centered simple waves for the pseudo-steady flow and give the expression of them.

3.1 Principal part of isentropic irrotational pseudo-steady centered waves

We discuss the properties of the principal part of the general centered simple wave for the system (2).

Definition 1     Let Λ₊(t) be an angular domain with the following boundaries (see Fig. 3):

(7)

A function (u, v, ρ)(ξ, η) is called a C₊ type centered simple wave solution for the system (2) with the origin (0, 0) as the center point if the following properties are satisfied^{[10-11, 14, 16]}:

(ⅰ) (u, v, ρ) can be determined by η=ξ tan α defined on a rectangular domain

as follows:

Moreover, (u₊, v₊, ρ₊) belongs to

(ⅱ) The function (u, v, ρ)(ξ, η) defined above satisfies Eq. (2) on Λ₊(t)\(0, 0).

(ⅲ) For any α∈[α₂, α₁], η=ξλ₊ gives the C₊ characteristic line passing through the origin (0, 0) with the slope tanα at the origin.

Substituting η=ξ tan α into Eq. (3), we obtain

where

and is called the principal part of this C₊ type centered simple wave.

Similarly, we have the following result.

Definition 2    Let Λ_(t) be an angular domain with the following boundaries (see Fig. 4):

(8)

A function (u, v, ρ)(ξ, η) is called a C_ type centered simple wave solution for the system (2) with the origin (0, 0) as the center point if the following properties are satisfied:

(ⅰ) (u, v, ρ) can be determined by η=ξ tan β defined on a rectangular domain

as follows:

Moreover, (u_, v_, ρ_) belongs to

(ⅱ) The function (u, v, ρ)(ξ, η) defined above satisfies Eq. (2) on Λ_(t)\(0, 0).

(ⅲ) For any β∈[β₂, β₁], η=ξλ_ gives the C_ characteristic line passing through the origin (0, 0) with the slope tanβ at the origin.

We get (u, v, c)(ξ, η)=(u_, v_, ρ_)(η, β) and the potential function

in the region

We define the principal part of the C_- type centered simple wave

and

3.2 Properties of the centered simple waves

In the following part, we give the properties of the C_± type centered simple waves.

Theorem 2 Assume that

and α₂≤ α≤ α₁ is the C₊ type centered simple wave solution of the system (2) in the pseudo-supersonic domain. Then, the principal part and satisfy

(9)

Proof From

(11)

substituting (u, v, ρ)(ξ, η)=(u₊, v₊, ρ₊)(ξ, α) into the system (6) and Eq. (3), we have

(11)

(12)

(13)

Since σ is the angle of the pseudo-velocity (φ_ξ, φ_η)=(U, V) and the positive ξ-axis, we have

(14)

By further computation, we obtain

(15)

Note (u₊, v₊, ρ₊, φ₊)∈ C¹(Λ₊(t)), and in the pseudo-supersonic flow region, and let ξ→ 0 in Eqs. (12), (13), and (15). Then, we have

(16)

(17)

(18)

Combining Eqs. (17) and (18), we have

According to the characteristic analysis in Section 2, for any point (ξ, η) on the C₊ characteristic line, we have

(19)

which is equivalent to (u₊(ξ, α)-ξ)sinα-(v₊(ξ, α)-ξ tan α)cosα=1 in the (ξ, α) plane. Let ξ→ 0. Then, we get and have the third equation of Eq. (9).

For the C_- type centered simple wave, we have the similar result.

Theorem 3   Assume that (u, v, ρ)(ξ, η)=(u_, v_, ρ_)(β, η), β₂≤ β≤ β₁ is the C_- type centered simple wave solution of the system (4) in the pseudo-supersonic region, where tanβ_i=λ_i (i=1, 2). Then, the principal part and satisfy

(20)

Proof   The proof of this theorem is similar to that of Theorem 2, and thus we omit the detail.

The following two theorems show that the C_± centered simple waves are determined by the principal part of the C_± centered simple waves.

Theorem 4   Assume that α₂≤ α≤ α₁ is the principal part of the C₊ centered simple wave, and the values of (u, v, ρ)(ξ, η) on the ray η=ξ tan α are defined as where tanα_i=λ_i (i=1, 2). Then, (u, v, ρ)(ξ, η) is the centered simple wave solution of Eq. (2) with the origin (0, 0) as the center point.

Proof    Firstly, from the third equation of Eq. (9), it is obvious that for any α₂≤ α≤ α₁, the straight is a C₊ characteristic line.

Secondly, we prove that (u, v, ρ)(ξ, η) satisfies Eq. (6). According to the definition, (u, v, ρ) is constant along the C₊ characteristic line. Therefore,

(21)

From the definition of (u, v, ρ)(ξ, η) and the second equation of Eq. (9), we obtain

(22)

At last, we have (u, v, ρ)(ξ, η) satisfying the pseudo-Bernoulli law.

(23)

By the definition of (u, v, c)(ξ, η) and Eqs. (8) and (9), we have

(24)

From Eqs. (23) and (24), we obtain that the Bernoulli law holds.

For the C_- type centered simple wave, we obtain the similar results.

Theorem 5    Assume that is the principal part of the C_- centered simple wave and the values of (u, v, ρ)(ξ, η) on the ray η=ξ tan β are defined as Then, (u, v, ρ)(ξ, η) is the centered simple wave solution of the system (2) with the origin (0, 0) as the center point.

4 Supersonic flow around the convex corner

Theorem 6    Assume that the density of the downstream flow is less than that of the supersonic oncoming flow (u₁, 0, ρ₁). Then, the oncoming flow turns the corner by an incomplete centered expansion simple wave R₊ and a constant state For any point (ξ, η) on the line η=ξ tan α, we have

(25)

where α₂ < α < α₁. The flow arrives at the constant state (u₂⁺, v₂⁺, ρ₂⁺) when

From

we get Therefore, we obtain the inverse function denoted by α=g(x).

Proof    Suppose the supersonic oncoming flow turns the corner by a C₊ type centered simple wave with the principal part Similar to constructing a simple wave for steady isentropic irrotational plane flow^[1], decomposing the pseudo-flow velocity

along the directions (-sinα, cosα) and (cosα, sinα), respectively, we have

where g (ξ, α)>0 in the pseudo-supersonic flow region. Let ξ→ 0. Then, we have

(26)

By direct calculation, we have

(27)

(28)

Differentiating Eq. (27), we obtain

(29)

Inserting Eq. (29) into Eq. (28), we have

(30)

Solving Eq. (30) with the initial data we have

(31)

Immediately, we obtain the expression of and in Eq. (25).

By virtue of the first equation in Eq. (9), there holds

Accordingly,

(32)

Differentiating Eq. (32) and combining Eqs. (30) and (31), we have

(33)

Using the oncoming flow (u₁, 0, ρ₁), we have

(34)

where α₁ is the Mach angle of the oncoming flow. From Theorem 5, we get that Eq. (25) defined in Theorem 6 is a C₊ type centered simple wave.

Secondly, we prove that Eq. (25) is a simple centered expansion wave, which is equivalent to proving where By a simple computation, we have

which means that and have the same sign for the C₊ type centered simple wave.

From Eq. (3), we have

Noting that the vector (ξ, η) is parallel to the direction of the C₊ straight characteristic line, we have by the second formula of Eq. (6). Therefore,

and we have

From Eq. (9), we have

which leads to Hence, the C₊ type centered simple (25) is an expansion wave.

From the last equation of Eq. (25), it is obvious that no cavitation will appear. From Eq. (25), we know that there must exist a unique constant α₂∈(-θ, α₁) such that

which implies that the oncoming flow turns the corner by an incomplete centered expansion wave defined on α∈[α₂, α₁] and a constant state

We describe the geometric procedure for isothermal flow in some more detail. Draw the epicycloidal arc Γ_ through the point q₁=(u₁-0, v₁-0)=(u₁, 0) in the plane (see Fig. 5). On Γ_, the point A' at the distance q₁ from O' corresponds to the zone of the constant state (u₁, 0, ρ₁). For any point P' on Γ_, draw O'P' parallel to the stream line The straight Mach line C₊ through O is determined as the line perpendicular to the direction of Γ_ at P'. The arc of Γ_ represents the incomplete simple wave, from which the flow emerges, parallel to the straight wall OB with the speed equal to the length of the segment O'B'.

When the supersonic or sonic downstream density is greater than that of the oncoming flow, the supersonic oncoming flow turns the convex corner by a centered compression wave.

Theorem 7    For the supersonic oncoming flow (u₁, 0, ρ₁), assume that the density of the downstream flow is greater than ρ₁ and Then, the oncoming flow turns the corner by a centered compression simple wave R_ and a constant state for any point (ξ, η) on the line η=ξ tan β,

(35)

where the expression of the centered compression wave in Eq. (35) is defined on (β₂, β₁) and

Proof    Suppose the oncoming flow turns the corner by a C_- type centered simple wave with the principal part where β is the characteristic angle of the C_- characteristic line η=ξ tan β. In the light of Theorem 5, we have

(36)

Similar to Theorem 6, decomposing the pseudo-velocity

along the direction (-sinβ, cosβ) and the C_- characteristic direction (cosβ, sinβ), respectively, we have

Let η→ 0. Then, we obtain

where corresponding to the pseudo-supersonic flow. Obviously,

(37)

(38)

Differentiating Eq. (38) with respect to β, we have

(39)

Substituting Eq. (39) into Eq. (36), we have

(40)

Solving Eq. (40) with the initial data we have

(41)

Immediately, we obtain the expression of and in Eq. (35).

By virtue of Eq. (17), there holds

Accordingly,

(42)

Then, we have

(43)

Using the oncoming flow (u₁, 0, ρ₁), we have

(44)

where is the Mach angle of the oncoming flow. From Theorem 5, we get that Eq. (35) defined in Theorem 7 is a C_- type centered simple wave.

Considering the demand of structuring a centered compression simple wave

we need Therefore, we obtain

From Theorem 5, Eq. (35) is the C_- type centered simple wave. Similar to the proof of Theorem 6, we can prove

Noting η>0, and sin(α-β)>0 in the pseudo-supersonic flow region, we can prove in pseudo-supersonic flow region, which means that the C_- type centered simple wave (35) is a compression wave.

In the light of Eq. (35), when

we have

which implies that the direction angle of (u, v) decreases from 0 to while β decreases from β₁ to β₂^*. Therefore, if the angle of the rigid wall OB satisfies there exists a unique constant β₂∈[β₂^*, β₁) such that which means that the oncoming flow turns the convex corner by a centered compression wave (35) defined on (β₂, β₁) and a constant state and flows along the rigid wall OB by the constant state (See Fig. 6(a)). Specially, when β₂=β₂^* and is a sonic state. When β₂∈(β₂^*, β₁) and is a supersonic state by the compressibility of the C_- type centered simple wave (35).

The geometric procedure for the centered compression wave (see Fig. 6) is similar to that of the incomplete centered simple wave in Fig. 5, and thus we omit the detail.

Combining Theorems 6 and 7, we obtain Theorem 1.

5 Conclusions

In this paper, the self-similar solutions for the 2D pseudo-steady isentropic irrotational supersonic isothermal flow around the convex corner are constructed. The supersonic flow turns the convex corner near the cusp of the corner locally by an incomplete centered expansion wave or an incomplete centered compression wave, depending on the conditions of the downstream state and the slope.