A-to-Z Guide to Thermodynamics,
Heat & Mass Transfer, and Fluids Engineering

Jets are liquid or gas flows in a space filled with a fluid with different physical parameters, namely, velocity, temperature, composition, etc. The distinctive features of these flows can be illustrated by gas jets flowing out into a space filled with gas at rest.

Figures 1 and 2 present two typical patterns of immersed jets. The first (Figure 1) corresponds to an isobaric jet in which static pressure is constant throughout an entire volume. The initial outflow velocity U0 can be either sub- or supersonic. The second pattern (Figure 2) is barrel-shaped in the initial section and corresponds to the supersonic efflux of an underexpanded jet when static pressure at the nozzle exit section pa substantially exceeds the environmental pressure pH. The pa/pH = n ratio is the most important parameter characterizing gas expansion at the initial section of the jet. Within one or a few "barrels" the external boundary of the jet contour becomes more regular and monotonically expands. Beyond this point, the jet pattern is similar to an isobaric jet (cf., Figures 1 and 2). The initial sections of length x0 of the two forms of jets differ from each other not only in external contours but also in the character of distribution of the most significant gas dynamic parameters. Figure 1. Subsonic isobaric jet. Figure 2. Supersonic jet.

The isobaric jet (Figure 1) has a core in the entrance region in which the axial gas velocity actually remains constant and equals U0. Between the core and the external boundaries there arises a dynamic, thermal, or diffusion boundary layer where turbulent mixing of jet particles and the quiescent medium occurs. Beyond the initial section x > x0, the boundary layers unite and axial flow velocity decreases inversely with x (the distance from the nozzle) for an axisymmetric jet, or with x−05 for a plane jet. The external boundary of the jet is not a tangential discontinuity, i.e., shows no jumps in its parameters and, like the thickness of a boundary layer, can be determined only approximately. If the surface on which gas velocity is 0.1 of that of axial flow is taken as the jet boundary, the thickness of the isobaric jet, in fact, grows linearly with x. For x > x0, the main section the proportionality factor is equal to 0.22, irrespective of jet shape, velocity and physical state. In the initial section (x < x0) the proportionality factor depends on the ratio of densities ρ0H or temperatures (T0/TH) (the subscripts "0" and "H" denote the initial jet and environmental parameters).

Figure 3 demonstrates the effect of preheating θ = T0/TH on the length of the initial section = x0/b0 for the axisymmetric (1) and plane (2) subsonic isobaric jets. Figure 4 shows the axial velocity for the axisymmetric subsonic jet as a function of preheating θ. Figure 3. Effect of preheating on the length of the initial section of a jet (1, axisymmetric; 2, plane). Figure 4. Axial velocity of a subsonic jet as a function of distance and preheat.

The characteristic property of the isobaric jet is that the transverse velocity components are negligible relative to the longitudinal velocity in any jet section. The profiles of excess velocity, temperature and impurity concentration in cross-sections of both immersed jet and the jet in the cocurrent flow are similar in shape. Figure 5 presents the velocity profiles at various cross-sections of a plane jet and Figure 6 shows a universal profile of all the sections in dimensionless coordinates. To describe the velocity profiles in the main section (x > x0) of the jet of any shape, Schlichting has derived the function f(η) Figure 5. Velocity distribution in a jet. Figure 6. Universal profile of velocity in a jet. Here η = y/b is the distance from the jet axis to the radius at a given cross-section b. The function f(η) holds for Ma0 << 1 and for small differences in density between the jet ρ0 and the environment ρH, i.e., at β = ρH0 ≈ 1. However, experiments show that the universal nature of the velocity profiles still holds in the range 0.25 ≤ β ≤ 4. For cross-sections in the main section (x > x0) the dependence of excess temperature on excess velocity is is valid. Here, Prt is the turbulent Prandtl number proportional to the ratio of heat released as a result of turbulent friction to that removed by turbulent mixing.

Experiments carried out at β = 0.03-300 show that Prt = 0.8 for axisymmetric jets and Prt = 0.5 for plane jets.

In jets flowing out of rectangular nozzles, flow sections can be distinguished in which the jet properties approach the appropriate analogous jets: the plane jet, near the nozzle and the axisymmetrie jet, at long distances. Between them, the jet is specified by three coordinates. Initially, jet expansion toward the minor nozzle axis is more intense than toward the major axis, and at some distance from the nozzle the jet pattern changes orientation by 90°, but the ratio of the larger to the smaller side turns out to be below the original one. Depending on outflow conditions, the jet may become an axisymmetric one at the distance of l00 h and over (h is the height of the shorter side of the nozzle). The mechanisms of velocity decay become the same as in an ordinary axisymmetric jet at different distances from the nozzle exit section in accordance with the nozzle aspect ratio. This characteristic distance can be estimated from experimental data using the dimensionless complex x/ > (3-6) (F0 is the total area of the nozzle exit).

In supersonic, underexpanded (n = (pa/pH) > 1) axisymmetric jets the entrance region is characterized by an intricate flow pattern. With excessive pressure, the jet first expands and its velocity grows. Near the nozzle edge, there arises the beam of rarefaction waves that facilitate gas expansion in the jet from pressure pa at the nozzle exit section to the pressure pH of the ambient gas. Acceleration of flow involves the Prandtl-Meyer expansion, and the jet boundary remains linear until it crosses the first characteristic. Overexpansion due to radial gas effusion gives rise to an intercepting shock 1 (Figure 2). Depending on the inclination angle of the shock (which, in turn, depends on the pressure ratio at the nozzle exit section), the intercepting shock is either reflected at a certain point lying on the axis (regular reflection) or forms a Mach disk (2 in. Figure 2). The Mach disk is the surface of a high intensity shock normal to the flow direction and interacting with the intercepting shock and the reflected shock wave at the triple point. The distance 1 from the nozzle exit section to Mach disk is proportional to the nozzle diameter da, the Mach number Maa, and the square root of the pressure ratio n. The reflected shock wave propagates in the outer region of the jet, making the gas flow once again from the center towards the periphery. When the reflected shock wave reaches the jet boundary the cycle is completed. Side oscillations may repeatedly occur along the flow before they damp altogether. The results from numerical methods demonstrate that at some length from the nozzle exit section, the flow is in good agreement with the model of gas expansion from a point source into vacuum.

Interaction of supersonic, underexpanded jet with a plane obstacle can be conventionally represented by three types of flow in accordance with the position of the nozzle relative to an obstacle. In the case when the obstacle is between the nozzle exit section and the point H ≤ 0.9xm, a stable interaction regime is observed (xm is the distance from the nozzle exit section to the Mach disk in the case of a free, supersonic jet). As the distance from the obstacle to the nozzle exit section increases in the (0.9-1.3)xm interval, a peripheral supersonic flow, possessing a high stagnation pressure, pinches the central subsonic flow to give rise to a peripheral stagnation point. Closure of the central zone of flow brings about accumulation of gas in it and a shift of the central shock toward the nozzle, which increases pressure before the obstacle with subsequent opening of this zone, etc. With a further increase in the distance from the obstacle to the nozzle exit section, the flow is stabilized and a stable circulation zone is formed around the obstacle.

If the obstacle is positioned in the interval (1.8-2.1)xm, an oscillation of the nozzle section wave structure arises once again, but the intensity of oscillations and the noise generated are much weaker than in the first instability regime. As the distance from the obstacle increases, the first barrel of the jet becomes undisturbed and a central compression shock, resulting from interaction of the second barrel with the obstacle, appears before the obstacle. The boundaries between the interaction regimes indicated above hold in the range n = 2-20, Maa =2-3.

The changes in heat transfer characteristics from the jet to the obstacle in the above cases are considered next. If the distance is short < 1H ( 1H is the distance from the nozzle exit section to the obstacle at which instability occurs) and the flow around the obstacle is stable, the maximum heat transfer coefficient a is at the stagnation point (curves 1-3, in Figure 7). With a greater distance from it α monotonically diminishes. As the pressure ratio increases and at = const, α profiles become fuller near the stagnation point, remaining virtually constant in the center of the obstacle and growing substantially at the periphery. It should be noted that with variation of Maa the α value and its character of distribution change only slightly. A further increase in within the stationary flow gives rise to a peripheral maximum α which shifts toward the stagnation point (curves 4-6 in Figure 7), as the obstacle is moved further from the nozzle. The same effect on α variation is exerted by n at = const. The peripheral maximum in α arises due to the effect of turbulent pulsations on heat transfer in the inner mixing zone, originating from the triple point of the central compression shock. The peripheral maximum is also characteristic of unsteady flow. Interaction beyond the first barrel results in only one maximum at the stagnation point. Figure 7. Variation of heat transfer coefficients with peripheral distance from stagnation point.

In the modeling of high-intensity, convective heat transfer in jets, determination of maximum Reynolds number at given values of pressure and temperature in a plenum chamber of a gas dynamic unit is of fundamental importance. This is reduced to an analysis of conditions under which the jets of finite diameter interact with blunt bodies of various shapes. It has been established that the maximum Re number, nonseparated flow around a hemispherical blunt body corresponds to the outflow regime at the nozzle section Maa = 2.5, the diameter of the body being no more than 0.9 of the nozzle diameter. The model can be enlarged about half as much again by using the cool gas wake with the same Mach number Maa = 2.5 at the nozzle exit section. The use of cool wakes makes it possible to sharply reduce the requirements for heat output of a test setup. However, all the conditions minimizing disturbances on the jet mixing boundary (shaping the nozzle edge, equalizing static pressures and Mach numbers in the exit section, smaller size of an edge) must be satisfied.

Short-duration and stationary jets differ greatly in both structure and parameter distribution.

A schematic representation of the initial stage of underexpanded, supersonic, short-duration jet is presented in Figure 8. Shock wave 1 appears in front of the jet gas, whereas secondary shock wave 2 is formed in the jet itself. Three kinds of toroidal eddies are formed, with different causes underlying their formation. A vortex ring 3 emerges as a result of pulsed enhancement of pressure in the nozzle exit section at the initial moment of efflux and resembles in nature annular Wood vortices. Vortices of type 4 are formed due to instability of the forward front of contact discontinuity. Vortex 5 is formed as a result of swirling of the turbulent shear layer, which originates along the lateral boundary of the jet, and, during subsequent development of the jet, takes place at its "head." This increases by two or three times the cross sectional dimensions of the jet as compared to a stationary jet with the same parameters. Figure 8. Schematic representation of underexpanded short-duration jet.

Just as in the case of the stationary jet, where the ratio of stagnation temperature to the environmental temperature and the Mach number may be such as to bring about quantitative changes in jet parameters, certain combinations of the Mach number and the temperature in short duration jets also cause structural changes in the flow. Around the jet, there may appear a region of backflow that is a swirling flow resembling a cocoon.

Figures 9 schematically depicts (a) the cold gas jet being formed, (b) the hot gas jet being formed, and (c) the stationary underexpanded jet. All three jets have identical pressure ratio. Figure 9. Transient behaviors of short-duration jets (a, cold gas, b, hot gas) compared with stationary jet (c).