14 Resonant, or standing, waves

Induction manifold design



499



the air is elastic and has mass, it responds by surging forward to restore the

pressure, thus initiating an alternating set of depression and compression

pulses, which are in fact sound waves travelling along the pipe at the speed

of sound.

At low speeds and with valve overlap, there can also be a slight puff-back

of gas from the combustion chamber into the inlet port, but this comes after

the initiation of the sound wave. It is less forceful and does not necessarily

significantly interfere with the resonance, though the larger the overlap, the

longer is the period available for such pulses and others from the exhaust

system to have an effect. The kinetic energy in the waves and momentum of

the flow increase with engine speed. This is because of the consequently

increasing depression in the cylinders, and therefore the pressure ratio across

the valve throat, with speed.

Resonant vibration phenomena are associated with mass–spring systems.

The mass is that of the column of air in the pipe and the spring element the

compressibility of that air. One wavelength λ is a complete cycle, or 2π

radians, and therefore is equal to L in the top diagrams in Figs 13.20 and

13.22, and 4L/3 in Fig. 13.21. The phase difference between the displacement

and pressure waves is always π/4, or 90°.

At this point, some clarification as to what exactly happens at the open

ends of a pipe is necessary. On reaching the open end remote from the valve,

a negative pressure wave sucks a slug of air in, and a positive pressure wave

propels a slug out. In both instances these effects take place against the

influence of atmospheric pressure, so there is an inertia-driven over-swing

followed by a bounce-back accompanied by a phase change.

If we plot the axial vibrations in the pipe to a scale such that the maximum

amplitude of displacement in each direction equals the radius of the pipe

section, they can be illustrated as shown in the top and bottom diagrams in

these illustrations. In each, the upper diagram represents the second-order,

and the lower one the first-order, or fundamental, mode of vibration.

λ = 4/3 L



A



B



A



C



B



λ = 4L



D



Fig. 13.21 Fundamental and first overtone

modes of vibration of air in a pipe one end of

which is closed and the other open



500



The Motor Vehicle

λ=L



A



B



C



D



E



A



B



C



D



E



λ = 2L



Fig. 13.22 Fundamental and first

overtone modes of vibration of air

in a pipe both ends of which are

open



From the upper diagram in Fig. 13.20, it can be seen that axial motion of

the air is positively stopped by the closed ends, A and E, of the pipe. These

ends are therefore displacement nodes. Mid-way between them is a third

displacement node, while B and D are displacement anti-nodes. Because the

air alternately moves towards and is bounced back from the displacement

nodes, A and C and E are pressure anti-nodes. In other words, while the

pressure remains constant at B and D, it fluctuates cyclically at A, C and E.

This condition can occur in an induction pipe only when both a throttle and

inlet valve are closed so, as regards manifold tuning, it is not of practical

significance but it is relevant for automotive engineers concerned with body,

cab or saloon noise.

If one end of the pipe is open, Fig. 13.21, the air at that end is free to be

displaced, so it becomes a displacement anti-node, which accounts for the

different arrangement of the displacement curves for the fundamental mode

of vibration and the overtones. This condition can arise when the inlet valve

is closed and the opposite end of the inlet pipe open.

For a pipe open at both ends, the fundamental and first overtone harmonics

are shown in Fig. 13.22. The third harmonic is illustrated in Fig. 13.23. Since

this is a condition that arises only when both the inlet valve and pipe end are

open, it is of significance in relation to resonance effects initiated by the

sudden pening of the inlet valve.

Clearly there must be some displacement beyond the open end before a

reflection can occur, so a correction factor has to be applied to the length of

the pipe. In fact, the effective length of an open end is L plus about 0.6 times

its radius r so, for one open and one closed end, the correction factor is L (1

+ 0.6r), and for a pipe with both ends open it is L (1 + 1.2r). The time t taken

for the completion of one wavelength is called the periodic time, or the

period of the vibration, and the time required for the pulse to return to the

inlet valve is 2L/c, where c is velocity of sound in the induction pipe. For



Induction manifold design



501



λ = 2/3L



A



B



C



D



E



F



G



Fig. 13.23 Third harmonic mode of vibration of air in a pipe with both ends open



several reasons, however, this is rarely directly applicable in the context of

induction-system tuning. First, the configurations of the ends of the passages

are not those of a plain pipe end; secondly, there are other influencing factors

such as air temperature and diameter of pipe; thirdly, and perhaps more

important, c is not constant for large-amplitude waves such as occur in

induction pipes. More accurate results can be obtained if a cyclical mean

value of c is used.



13.15



Pipe end-effects



Movement of the air into a pipe in general, and its displacement due to the

vibrations, tend to cause turbulence around its open end, reducing the efficiency

of flow. This adverse effect can be considerably reduced by flaring the open

end of the pipe to form a trumpet of approximately hyperbolic section, so

that it guides the air flow smoothly in and thus increases the coefficient of

inflow by up to about 2%. The effective length of a pipe with such an end

fitting is that of the parallel portion plus about 0.3 to 0.5 of the length of the

flare. If the outer ends of the pipes terminate in apertures in a flat plate, or

in the wall of a plenum chamber, their flares should not only extend well

clear of the flat surfaces but also be clear of any adjacent walls, to ensure that

the approach velocity is well below that within the pipe.

Tapering the pipe, increasing its diameter from the inlet port to its open

end, also reduces the end-effect. This is sometimes done on very high-speed

engines, for example in racing cars. The aim is to reduce the velocity of flow

into the open end, and therefore the tendency for turbulence to be generated

there. However, it is not conducive to the generation of powerful standing

waves. Incidentally, any reduction in the velocity of flow will also reduce the

viscous drag between the air stream and the walls of the tube.



13.16



Frequencies, wavelengths and lengths of pipes



From the four illustrations, it is easy to see that the harmonic frequencies for

pipes closed or open at both ends are f1, f2, f3, f4,…, fn, while those of the pipe

closed at one end and open at the other are the odd numbers, f1, f3, f5, f7,…,

fn. The formula from which these frequencies can be obtained is f = c /λ,



502



The Motor Vehicle



where c is the velocity of sound in air and λ is the wavelength. The

frequencies of the first three modes of vibration in each case therefore are as

follows—

Pipes with closed ends

One end open

Both ends open



f1 = c/2L

f1 = c/4L

f1 = c/2L



f2 = C/L

f2 = 3c/4 L

f2 = c/L



f3 = 3c/L

f3 = 5c/4L

f3 = 3c/2L



For waves of small amplitude the velocity of sound in dry air is √γp/ρ,

where p is the gas pressure, ρ is the density, and γ is the ratio of the specific

heats of the gas. At the standard temperature and pressure in free air, this

velocity becomes 331.4 m/s. Standard temperature and pressure is 298.15 K

and 105Pa (1 bar). Potential for some slight confusion arises, however, when

referring back to data predating the universal introduction of SI units because,

at the latter point, it became 273.15 K (0°C) and 101.325 Pa. At velocities of

more than Mach 0.25, viscous friction losses impair performance.

Whichever version of the speed of sound in free air is taken, it is independent

of frequency and, because pressure divided by density is constant, it can be

considered also to be independent of variations of pressure, certainly of the

magnitudes experienced in inlet manifolds. The velocity of sound varies

with temperature according to the following relationship—

cθ = c0 √(1 + αθ)

where cθ and c0 are the velocities of sound at θ and 0°C respectively, and α

is the coefficient of expansion of the gas. While the local velocity of sound

is dependent only on the temperature and composition of the gas, in induction

pipes it is influenced also by diameter, Fig. 13.24. This is because of the

effect of viscous friction between the gas and the walls of the pipe. Frequency

is also affected, but relatively slightly, by the length: diameter ratio and

internal smoothness of the pipe, both of which influence the degree of damping

of the flow.

Since γ is dependent on the nature of the gas, the presence of fuel vapour,

as in carburetted or throttle body injected spark ignition engines, will also

affect the speed of sound in the manifold. Even so, because extreme accuracy

of calculation is generally unattainable, except possibly where the system

comprises a set of straight tubes, this is not of much practical significance.

Indeed, induction systems have to be optimised experimentally, for example

by the use of telescopic elements, during development.

The amplitudes of the resonant pressure pulsations too are modified by

damping. This can be due to roughness of the inner faces of the walls of the

induction tract, the presence of bends, and obstructions such as throttle

valves and inlet valve stems and guide ends. From damped and undamped

resonance curves in Fig. 13.25, it can be seen that the effect of damping is

not only a reduction in maximum amplitude but also it rounds off the peak,

and spreads the resonance over a significantly wider range of frequencies.

In general, any bends in the pipes should be as close as practicable to the

inlet valve ports, blended smoothly into the straight sections, and their radii

should not be less than four times that of the bore of the pipe. This arrangement

leads to a minimum of both viscous losses and interference with the tendency

for the air in the pipe to resonate freely.