In thermoacoustic devices with a looped tube, it has been pointed out that acoustic streaming circulating the loop leads to unwanted heat loss. In this research,, we measured the acoustic streaming velocity in a double-loop thermoacoustic Stirling cooler, and show that the streaming circulating the loop accounts for a heat loss of 150 W when input heat to the prime-mover loop is 408 W. It is also shown that the installation of an elastic membrane effectively suppresses the heat loss and thereby increases the performance of the prime-mover loop and cooler loop.